Paper Title: Shift operators and their classification.















Shift operators and their classification
    Maria Carvalho  Partially supported by CMUP, member of LASI, which is financed by national funds through FCT – Fundação para a Ciência e a Tecnologia, I.P., under the projects with references UIDB/00144/2020 and UIDP/00144/2020, and also supported by the project PTDC/MAT-PUR/4048/2021. 
 
Udayan B. Darji Thanks partial support of CMUP and Jagiellonian University  as Jagiellonian Scholar.

Paulo Varandas 
Partially supported by the grant CEECIND/03721/2017 of the Stimulus of Scientific Employment, Individual Support 2017 Call, awarded by FCT.


    August 5, 2024
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§ ABSTRACT  
We introduce a class of linear bounded invertible operators on Banach spaces, called shift operators, which comprises weighted backward shifts and models finite products of weighted backward shifts and dissipative  composition operators. We classify vast families of these shift operators, 

and this classification yields verifiable conditions which we use to construct concrete examples of shift operators with a variety of dynamical properties. As a consequence, we show that, for large classes of shift operators, generalized hyperbolicity is equivalent to  the shadowing property.
[2020 Mathematics Subject Classification: Primary 37C50, 47B37, 37B65; Secondary 47B33, 47B01, 

 47A15, 
 
 37C60, 
 
 37D99,
 
 47B80
 
 
Keywords: Shift operators, weighted shifts, dissipative operators, generalized hyperbolicity, shadowing]







§ INTRODUCTION

Linear dynamics is the study dynamical properties of linear operators on infinite dimensional Banach spaces. One of its aims is to classify dynamical properties of linear operators. It is often a non-trivial problem to decide, given a fixed Banach space X, if it indeed admits linear operators with dynamical properties such as transitivity, mixing, chaoticity, frequent hypercyclicity, non-trivial invariant measures, etc. For example, it was shown by Ansari <cit.> and Bernal <cit.> that every infinite dimensional separable Banach space admits a mixing operator. This was generalized to Frećhet spaces by Bonet and Peris, using a key idea of Salas <cit.> which states that transitivity persists under perturbation of the identity by weighted backward shifts.



It is also an important problem to determine whether well-known results of classical dynamics in the setting of compact metric spaces hold in the setting of Banach spaces. For example, one can construct concrete examples which distinguishes the dynamical properties of transitivity, weak mixing, mixing and chaoticity by considering weighted backward shifts. In fact, one characterizes weighted backward shifts with a given dynamical property and then it is simply a matter of choosing proper weights to distinguish these properties. 



The class of weighted backward shifts is also dispensable when studying hyperbolic linear dynamics. For example, what is the relationship between hyperbolicity and the shadowing property? It is well-known that, in finite dimensional Banach spaces, hyperbolicity is equivalent to the shadowing property <cit.>. However, it was an open problem whether the shadowing property implies hyperbolicity in infinite dimensional spaces. This was settled negatively <cit.> by constructing a weighted backward shift with such properties. In the process, a new notion of hyperbolicity arose. This notion was extracted from <cit.> and named generalized hyperbolicity. It turns out that generalized hyperbolicity implies shadowing. However, the converse remains open. A characterization of weighted backward shifts with the shadowing property was given in <cit.>, essentially yielding that for weighted backward shifts, the shadowing property is equivalent to generalized hyperbolicity. It is also known that hyperbolicity is equivalent to expansivity and the shadowing property <cit.>.



One can infer from the above that weighted backward shifts are essential in the study of linear dynamics, in the same spirit as symbolic dynamics is essential to the study of topological dynamics and ergodic theory. Despite their immense utility, weighted backward shifts have their shortcomings. For example, weighted backward shifts on ℓ_p spaces fail to distinguish between chaoticity and frequent hypercyclicity <cit.>. To remedy this, they have been generalized in various directions. One of them is a recent investigation of composition operators on L^P(X, μ), also known as Koopman operators, induced by a bimeasureable map f L^P(X,μ) → L^P(X,μ). More precisely, when the Radon-Nikodym derivative d  (μ∘ f)/d μ is bounded below, the map T_f L^p(X, μ) → L^p(X, μ) given by T(φ) = φ∘ f is a bounded invertible linear operator. This class of composition operators captures weighted backward shifts and numerous results which characterize dynamical properties of T_f have been established. For example, it has been characterized when T_f is transitive or mixing <cit.> or T_f Li-Yorke chaotic <cit.>. Under the assumption of f being dissipative and satisfying a condition called bounded distortion, it is also possible to characterize when T_f is chaotic, frequently hypercyclic <cit.> and having the shadowing property <cit.>. In fact, under those two assumptions, 

one can show that T_f admits a particular weighted backward shift B_ω as a factor. Moreover, T_f has one of numerous dynamical properties if and only if B_ω does <cit.>.




Another important and well-studied generalization of weighted backward shifts on ℓ_p is when the underlying space is more general, for example a Köthe space or a Fréchet space. Very recently, an intriguing study has been taken up where weighted shifts on directed trees are investigated <cit.>.





In this article, we generalize weighted backward shifts in a different direction. We start with a Banach space X and a sequence of suitable linear invertible bounded operators (S_n)_n ∈ on X. We study the weighted shift induced by such operators. Precise definitions are given in Section <ref>. In particular, we show that several of the well-studied operators, such as dissipative Koopman operators, fall under the umbrella of our setting. It was shown <cit.> that a shift-like operator admits a weighted shift as a factor; moreover, it was proved that such an operator satisfies a given property 𝒫 if and only if its weighted backward shift factor does, where 𝒫 is one of numerous properties, such as shadowing, expansivity, transitivity, mixing, etc. We will show that our framework also includes this class of operators, meaning that all shift-like operators are conjugate to shift operators.




We classify vast families of these shift operators, including the ones generated by orthogonal matrices, diagonalizable matrices, rotation matrices and hyperbolic matrices. This classification yields verifiable conditions which we use to construct concrete examples of shift
operators with a variety of dynamical properties. 




As stated earlier, it is still an open question whether the shadowing property implies generalized hyperbolicity for bounded invertible linear operators. We show such is the case for a large class of shift operators. 



We would like to point out some  recent developments related to our work. Menet and Papathanasiou investigate weighted shifts on ℓ_p-sums and c_0-sums <cit.>. Bernardes and Peris <cit.> carried out a detailed study of various types of shadowing for operators defined on Banach spaces and Fréchet spaces. In particular, they showed that finite shadowing and shadowing are not the same for operators on Fréchet spaces, although they are equivalent for operators on Banach spaces. Actually, the latter turns out to be more difficult to prove than in the setting of compact spaces. It is still not obvious what the definitions of expansivity and hyperbolicity should be for operators on Fréchet spaces. An investigation along this direction is carried out by Bernardes, et al <cit.>.



The paper is organized as follows. In Section <ref>, we introduce the basic terminology and state the main mathematical contributions and their consequences. In Section <ref> we give applications of our results, including some concrete examples which show how to construct operators with or without certain dynamical properties. Section <ref> consists of the proofs of the main results. Section <ref> states some open problems of interest. 



 

§ MAIN RESULTS


In what follows, given 1 ≤ p < +∞ and a vector space Y over the field = or , we denote by ℓ_p(Y) the space of bilateral sequences of elements of Y endowed with the p-norm.


Let X be a Banach space and 𝒮 = (S_n)_n ∈ be a sequence of linear bounded invertible operators on X for which there exists C > 0 such that 

    max_n  ∈  {S_n,  S_n^-1}  <   C.

Given a Banach space B⊂ X^ℤ, the shift operator on B generated by 𝒮 is the linear bounded invertible operator σ_𝒮 B→ B defined by 

    σ_𝒮((x_n)_n  ∈ )  =  (S_n+1(x_n+1))_n ∈ .

This class of operators includes the weighted backward shift. For example, when X = and S_n→ is given by S_n(x) = ω_n x for a sequence ω = (ω_n)_n  ∈ which is bounded and far away from zero, then we obtain the weighted backward shift that we denote by B_ω. 

 

 §.§ General properties


In this section we start illustrating the relevance of the shift operators by describing two classes of linear operators which are modeled by them.



Let (E_n)_n ∈ ℤ be a collection of closed subspaces of a Banach space X. We say that (E_n)_n ∈ ℤ is a dissipative decomposition of X if, given x∈ X, for every n ∈ℤ there is x_n∈ E_n such that

 
  (a) the sums x^+ = ∑_n=0^+∞ x_n and x^- = ∑_n=-∞^-1 x_n converge;
 
  (b) x = x^-+x^+ and this representation is unique;
 
  (c) for every n∈ℤ, the projection p_n X → E_n, given by p_n(x) = x_n, is continuous.
 

When the previous conditions are valid we write X = span (⊕_n ∈ ℤ E_n). A linear operator T X→ X is said to be dissipative if there exists a closed subspace E_0⊂ X such that the collection
(T^n(E_0))_n ∈ ℤ is a dissipative decomposition of X. 


Dissipative decompositions of Banach spaces appear naturally in the context of linear dynamics. 
This is the case of weighted backward shifts on ℓ_p-spaces, for 1 ≤ p < +∞ (cf. <cit.>), shift-like operators (cf.
<cit.>) and generalized hyperbolic operators on Hilbert spaces with a dense set of periodic orbits (see <cit.>).



Suppose that T X→ X is a dissipative bounded invertible linear operator on a Banach space X. Then T is conjugate to a shift operator σ_S B →B, where B is a Banach space, 𝒮 = (S_n)_n ∈ ℤ and S_n is the identity map for every n ∈ℤ. 



The proof of Theorem <ref> does not provide, in general, an explicit description of the Banach space where the shift operator is defined. However, under additional assumptions or within specific frameworks, we may succeed in describing the underlying space. One such framework is the class of dissipative composition operators that were introduced in <cit.>. It can be easily verified that the definition of dissipative composition operator given below is consistent with our definition of dissipative operators defined above. 


Let f M→ M be a bijective bimeasurable map on a σ-finite measure space (M, 𝔅, μ) such that 
the Radon-Nikodym derivative dμ f/dμ exists and is bounded from below and from above. This assumption guarantees that the Koopman operator T_f induced by f, acting on L^p(μ), for some 
1 ≤ p < +∞, by

    φ∈ L^p(μ)     ↦     T_f(φ) = φ∘ f

is well-defined, invertible and  T_f^-1 = T_f^-1.
In case M = ⋃̇ _k=-∞^+ ∞ f^k (W) for some  W ∈𝔅 with 0 < μ (W) < +∞, the operator T_f is called a dissipative composition operator.  


We note that the next theorem is not a corollary of the previous one as we obtain B as a specific L^p space.


A dissipative composition operator T_f L^p(μ) → L^p(μ), associated with a dissipative system (M,𝔅,μ, f) generated by a set W, is conjugate to a shift operator σ_𝒮 B→ B, where 

    B = { (ψ_n)_n ∈ ℤ∈L^p(μ|_W)^ℤ∑_n ∈ ℤ ∫_W |ψ_n|^p  dμ f^n/dμ  dμ < +∞}
 
endowed with the norm 

    (ψ_n)_n ∈ ℤ_ B = (∑_n ∈ ℤ ∫_W |ψ_n|^p  dμ f^n/dμ  dμ)^1/p


and 𝒮=(S_n)_n ∈ ℤ, with S_n being the identity map on L^p(μ|_W) for every n ∈. 



It is worth mentioning that, even though all the operators S_n are the identity map, the shift operator σ_𝒮 can exhibit a variety of dynamical behaviors. For instance, depending on the dissipative system (M,𝔅,μ, f), the shift operator σ_𝒮 may or may not have the shadowing property, be transitive, be expansive, etc (cf. <cit.>). 




 §.§ Factors of shift operators


In what follows we will find weighted backward shifts defined on ℓ_p(), for ≤ p < +∞, as factors of shift operators. To this end, we introduce some notation which expedite the  statements of the main results. 

Consider a Banach space X and a sequence (S_n)_n  ∈ of bounded invertible linear operators on X. For n, m ∈, let

    S_[n,  m] = [ S_n   ∘ ⋯ ∘  S_m        if n  < m;              S_n        if n  = m;               Id        if n  > m ].

Given x∈ X∖{0}, define the sequence of weights ω(x) = (ω_n(x))_n ∈ ℤ∈ℝ^ℤ by 


    ω_n(x)  = [ ( S_[1,n-1] )^-1(x)/( S_[1,n] )^-1(x)                            if   n > 0;               S_[n,0](x)/S_[n+1,0](x)                           if   n ≤ 0. ]











Since are assuming that there exists C > 0 such that 

    max_n  ∈  {S_n,  S_n^-1}  <  C

we are sure that the shift operator σ_𝒮 is a well-defined bounded invertible operator acting on ℓ_p(X), 1≤ p <+∞, and that the sequence (ω_n(x))_n ∈ defined by (<ref>) is bounded from below by C.



We shall now address a special class of sequences 𝒮=(S_n)_n ∈ ℤ.


Assume that X is a Hilbert space. A sequence 𝒮=(S_n)_n ∈ ℤ of bounded invertible linear operators on X has an orthogonal basis  if {e_n(x) x ∈} is orthogonal for every n ∈, where 

    e_n(x)  = [ ( S_[1,n] )^-1 (x)/( S_[1,n] )^-1 (x)                            if   n ≥ 0;           S_[n+1,0] (x)/S_[n+1,0] (x)                             if   n <0 ].






We note that, by construction, if S_n=L for all n ∈, then (e_n(x))_n ∈ is the normalized orbit of x ∈ by L. Moreover, in general, one always has 

    S_n+1 (e_n+1(x)) = ω_n+1(x) e_n(x)      ∀  n ∈ Z     ∀  x ∈.





We observe that, if S_n is orthogonal for every n ∈, then any orthogonal basis  of X is an orthogonal basis for the sequence 𝒮=(S_n)_n ∈ ℤ. In this case, it is known that each S_n is a scalar multiple of an isometry, say S_n = α_n I_n, with α_n∈∖{0} and I_n being an isometry. Thus, ω_n(x)=|α_n| for every n∈ and x ∈ X∖{0}. 


Let X be a Hilbert space and suppose that the sequence of operators 𝒮=(S_n)_n ∈ ℤ on X has an orthogonal basis . Then, for any x ∈, the weighted backward shift B_ω(x)ℓ_p( 𝕂) →ℓ_p(𝕂) is a factor of the shift operator σ_𝒮ℓ_p( X) →ℓ_p( X).


The following consequence of Theorem <ref> is due to the fact that, in general, factors preserve transitivity, mixing 
and the shadowing property. 


Suppose we are in the setting of Theorem <ref>. If σ_𝒮 is transitive, mixing or has the shadowing property, then the same holds for B_ω(x) for all x ∈.



We note that, according to  Corollary <ref>, while proving that σ_𝒮 does not have a property in question (transitive, mixing or shadowing), we may simply find x ∈ such that B_ω(x) does not have the corresponding property (see Example <ref>).





 §.§ Classification theorems

We next describe the shift operators σ_𝒮 when X is a finite dimensional Hilbert space and 𝒮 = (S_n)_n  ∈ has an orthogonal basis. Given a product of a finite number of copies of the Banach space ℓ_p(𝕂), endow the product vector space with the sum norm.


Let X be a finite dimensional Hilbert space and assume that 𝒮=(S_n)_n ∈ has an orthogonal basis . Then, 
 for each 1≤ p <+∞,  
 σ_𝒮ℓ_p( X) →ℓ_p( X) is conjugate to the finite product of weighted backward shifts 
 
    ∏ _x  ∈  B_ω(x)  (ℓ_p(𝕂))^ X  →  (ℓ_p(𝕂))^ X

 where X stands for the dimension of X.



The reader may wonder why the weights ω_n are real numbers despite the fact that the underlying field  may be . Actually, it is well-known that every weighted backward shift on  with complex weights is conjugate to a weighted backward shift with positive weights <cit.>. This also follows from our Theorem <ref>. Moreover, every weighted backward shift with positive weights over  can easily be seen as conjugate to the product of two weighted backward shifts with positive weights over .


The assumptions on the previous theorem may seem restrictive. However, below we state an alternative classification theorem for the shift operators σ_𝒮 when X is a finite dimensional Banach space and orthogonality is replaced by uniform boundedness of projections. Let us introduce the necessary notions.



Given a Banach space X and a vector x∈ X, let F_x⊂ X denote the subspace spanned by x. Consider a basis  of X, x ∈ and the projection Π_x X→ F_x given by

    Π_x(∑_b ∈  α_b   b 
    ) = α_x  x

for any sequence of scalars (α_b)_b ∈ in 𝕂. For every x∈, the map Π_x is well defined, since every vector in X has a unique representation in terms of vectors in ; furthermore, Π_x is linear, bounded and Π_x∘Π_x = Π_x.



Given a sequence 𝒮 = (S_n)_n ∈ of bounded invertible linear operators on a Banach space X, we say that a basis  of X has 𝒮-bounded projections if there exists C>0 such that 

    sup_n ∈  sup_b ∈ _n  Π_b ≤ C  <  +∞,

where _n = {e_n(x) x∈}.




Let X be a finite dimensional Banach space, 𝒮=(S_n)_n ∈ ℤ be a sequence of operators on X and  be a basis of X with 𝒮-bounded projections. Then, for each 1 ≤ p < +∞, the shift operator σ_𝒮ℓ_p(X) →ℓ_p(X) is conjugate to the finite product ∏ _x  ∈   B_ω(x) (ℓ_p())^ X→ (ℓ_p())^ X.


Theorem <ref> allows us to strengthen Theorem <ref> by replacing orthogonality with suitable uniform lower and upper bounds on the angles between distinct vectors in each basis _n. 




Let X be a finite dimensional Hilbert space with dimension X ≥ 2 and assume that 𝒮=(S_n)_n ∈ has a basis  of X for which there exists 0 < γ < 1/(d-1) such that, for every n ∈, the angle ∡(u,v) between distinct vectors u,v ∈_n satisfies the condition

    cos∡(u,v)  ∈  [-γ, γ].
 
Then, given 1 ≤ p < +∞, the shift σ_𝒮ℓ_p(X) →ℓ_p(X) is conjugate to the finite product of weighted backward shifts ∏ _x  ∈  B_ω(x)(ℓ_p(𝕂))^ X→(ℓ_p(𝕂))^ X. 




We observe that, as d goes to infinity, the requirement in the previous corollary gets closer to orthogonality.



The next corollary provides another way of improving Theorem <ref>, by using Theorem <ref> and a new sufficient condition (which does not depend on the dimension d of the Hilbert space X) for a sequence 𝒮 = (S_n)_n ∈ to have 𝒮-bounded projections. Given a Hilbert space X, v ∈ X∖{0} and a subspace F⊂ X, consider the following infimum 

    ∡ (v,F)   =  inf_u ∈  F∖{0} ∡ (v,u).

For a basis  of X and v ∈, we denote by F_n,v the subspace of X generated by the vectors {e_n(b)  b ∈∖{v}}.


Let X be a finite dimensional Hilbert space with dimension d ≥ 2 and assume that 𝒮 = (S_n)_n ∈ has a basis  of X satisfying

    inf_n ∈ ℤ,    v  ∈   ∡ (e_n(v),F_n,v)   >   0.

Then, for every 1 ≤ p < +∞, the shift σ_𝒮ℓ_p( X) →ℓ_p( X) is conjugate to the finite product of weighted backward shifts ∏ _x  ∈  B_ω(x)(ℓ_p(𝕂))^ X→(ℓ_p(𝕂))^ X. 






 §.§ Equi-properties


It is known that some relevant properties of linear invertible bounded operators may be conveyed to factors. We will strengthen such results in the setting of Theorem <ref>.


    Let X be a Banach space and 𝒞 be a class of operators acting on X. We say that 𝒞 is:

    
  (a) Equi-transitive if for any non-empty open sets U,V⊂ X there exists n ∈ so that T^n(U)∩ V ≠∅ for every T∈𝒞.
    
  (b) Equi-mixing if for any non-empty open sets U,V⊂ X there exists N ∈ so that T^n(U)∩ V ≠∅ for every T∈𝒞 and every n≥ N.
    
    
  (c) Equi-shadowing if there exists K >0 such that, for all T∈𝒞 and any bounded sequence {z_n}_n  ∈ in X, there is a bounded sequence {x_n}_n  ∈ in X satisfying 
    
    x_n+1-T(x_n) = z_n     and    sup_n ∈  x_n < K sup_n  ∈  z_n.

  


Observe that, in case 𝒞 consists of a single operator T, equi-property 𝒫 is the same as property 𝒫 for T. Also notice that, if 𝒞 is finite, then equi-shadowing is equivalent to shadowing for all elements of 𝒞.



In what follows 𝒫 will stand for any of the properties summoned in Definition <ref>, that is, transitivity, mixing or shadowing.



Let X be a finite dimensional Banach space, 𝒮=(S_n)_n ∈ ℤ be a sequence of operators on X and  be a basis of X with 𝒮-bounded projections. If 
σ_𝒮ℓ_p( X) →ℓ_p( X) has property 𝒫, then 𝒞={B_ω(x) x∈} has equi-property 𝒫.


Applying Corollary <ref>, together with   Theorem <ref> and the fact that mixing, shadowing and generalized hyperbolicity are preserved under finite products and by linear conjugation, and letting 𝒬 be one of these properties, we conclude that:


Let X be a finite dimensional Banach space, 𝒮=(S_n)_n ∈ ℤ be a sequence of operators on X and  be a basis of X with 𝒮-bounded projections. Then the shift σ_𝒮ℓ_p( X) →ℓ_p( X) has property 𝒬 if and only if B_ω(x) has property 𝒬 for all x ∈.


It is easy to verify that Corollary <ref> fails for the shadowing property in case X is infinite dimensional. For instance, let X=ℓ_2(ℝ), = {e_j  j ∈} be the canonical basis of X and Tℓ_2(ℝ) →ℓ_2(ℝ) be defined by 
 
    T(e_j)  = (1+1/2^j)  e_j

 for all j ∈. Then the shift operator σ_𝒮ℓ_p(X) →ℓ_p(X),  determined by 𝒮=(S_n)_n ∈ ℤ with S_n = T for every n ∈, does not have the shadowing property (cf. <cit.>), although for all x ∈ the weighted backward shift B_ω(x)ℓ_p() →ℓ_p() has it.




 §.§ Shadowing vs. generalized hyperbolicity





 
  

 

It is known that generalized hyperbolic linear operators satisfy the shadowing property 
(cf. <cit.>). On the other hand, a  weighted backward shift on ℓ_p(𝕂) has the shadowing property if and only if it is generalized hyperbolic (cf. <cit.>); and a finite product of generalized hyperbolic operators is clearly generalized hyperbolic. Since, under the hypothesis of Theorem <ref>, 
σ_𝒮 is conjugate to a finite product of weighted backward shifts on ℓ_p(𝕂), we obtain the following equivalence:

 
Let X be a finite dimensional Banach space, 𝒮=(S_n)_n ∈ ℤ be a sequence of operators on X and  be a basis of X with 𝒮-bounded projections. Then the shift σ_𝒮ℓ_p(X) →ℓ_p(X) is generalized hyperbolic if and only if it has the shadowing property.





 §.§ Sufficient conditions for shadowing on shift operators


Applying Bernardes-Messaoudi characterization of the weighted backward shifts with the shadowing property (cf. <cit.>) to the setting of Corollary <ref>, and rewriting w_n(x) in terms of the S_n's, we have the following immediate criterion for shift operators.


Let X be a finite dimensional Hilbert space and assume that the sequence 𝒮=(S_n)_n ∈ ℤ has a basis  with 𝒮-bounded projections. Then σ_𝒮ℓ_p( X) →ℓ_p( X) has the shadowing property if and only if  for every x ∈ℰ one of the following conditions holds: 


  (A) 

    lim_n → +∞(
                    max{sup_k ∈ S_k-1^-1 … S_1^-1 (x)/S_k+n^-1 … S_1^-1 (x),
            sup_k  ≥  n (S_-k …  S_0(x)/S_-k+n …  S_0 (x)), 
            sup_0 ≤  k  < n (S_-k …  S_0(x)/S_-k+n^-1 …  S_1^-1 (x))
            })^1/n <1.

        
  (B) 
        
    lim_n → +∞(
                    max{inf_k ∈ S_k-1^-1 … S_1^-1 (x)/S_k+n^-1 … S_1^-1 (x),
            inf_k  ≥  n(S_-k …  S_0(x)/S_-k+n …  S_0 (x)),
            inf_0 ≤  k  < n(S_-k …  S_0(x)/S_-k+n^-1 …  S_1^-1 (x))
            })^1/n >1.

            
        
  (C) 
        
    lim_n → +∞  sup_k ∈  (S_-k-n …  S_0(x)/S_-k+1 …  S_0 (x))^1/n < 1
                  and      lim_n → +∞  inf_k ∈ (S_k-1^-1 … S_1^-1 (x)/S_k+n^-1 … S_1^-1 (x))^1/n > 1. 
             aaaaaaaa
 





A shift operator σ_𝒮 may fail to satisfy the shadowing property even though each of the operators S_n in the sequence 𝒮 has that property, as the next example illustrates. 

[Operators without shadowing]
Consider the Banach space ℝ^2 endowed with the Euclidean norm and 
ℓ^∞(ℝ^2)  = {(x_n, y_n)_n ∈(ℝ^2)^ℤ sup_n ∈ ℤ (x_n, y_n) < +∞}.
Let Tℝ^2 →ℝ^2 be the linear invertible bounded operator given by 

    (x,y)∈ℝ^2     ↦     T(x,y) = (2x, 1/2 y).

The map T is hyperbolic, hence satisfies the shadowing property. Consider now the shift operator σ_𝒮 where 𝒮=(S_n)_n ∈ ℤ is defined by S_n=T if n is odd and S_n=T^-1 otherwise. This shift operator does not have the shadowing property in view of Corollary <ref>.







 

§ APPLICATIONS


To illustrate the scope of Theorems <ref> and <ref>, in this section we will discuss several examples of shift operators exhibiting a variety of dynamics.



 §.§ Sequences of operators with an orthogonal basis


[Rotation operators]
Consider a rotation matrix in X = ℝ^2 given by

    R_θ =
    [  cos (2πθ) -sin (2πθ);  sin (2πθ)  cos (2πθ) ]

where θ∈  ]0,1[. Let = (S_n)_n  ∈ where S_n = 1/2 R_θ_n for all n ∈ and θ_n ∈  ]0,1[. Then any orthogonal basis of ^2 is an orthogonal basis for . By Theorem <ref>, we know that σ_ℓ_p(ℝ^2) →ℓ_p(ℝ^2) is conjugate to the product of weighted backward shifts B_ω× B_ω, where ω=(w_n)_n ∈ is given by ω_n=1/2 for every n∈.
From Corollary <ref>
we also conclude that σ_𝒮 has the shadowing property, and is in fact hyperbolic. 




[Diagonal operators]
Consider d ∈, X = 𝕂^d and invertible diagonal matrices
        
    S_n=
      [ λ_n(1)      0      …      0;      0 λ_n(2)      …      0;      ⋮      ⋮      ⋱      0;      0      0      … λ_n(d) ].
 
Clearly, the canonical basis of ^d is an orthogonal basis for the sequence = (S_n)_n ∈. By Theorem <ref>, the shift σ_ℓ_p(𝕂^d) →ℓ_p(𝕂^d) is conjugate to a finite product of weighted backward shifts. Moreover, σ_ has the shadowing property if and only if (λ_n(t))_n ∈ satisfies Bernardes-Messaoudi criteria <cit.> for every 1 ≤ t ≤ d.
Furthermore, σ_𝒮 has the mixing property if and only if 
(λ_n(t))_n ∈ satisfies the criteria in <cit.>.



We note that, conversely, a product of d weighted backward shifts, each defined in ℓ_p(𝕂), is conjugate to the shift operator σ_ generated by the sequence (S_n)_n ∈ of diagonal matrices, with respect to the canonical basis in 𝕂^d, whose entries are precisely the weights of the factors.


[Operators with an orthogonal basis of eigenvectors] Let X = ℝ^2 and consider 

    L =
      [ 2 1; 1 1 ].

The matrix L is not orthogonal, yet it is diagonalizable with eigenvalues 3±√(5)/2 and eigenvectors = {(1,√(5)-1/2),  (1,-√(5)+1/2)}, which are orthogonal. Thus, the sequence 𝒮=(S_n)_n ∈ ℤ of operators in ℝ^2 given by S_n=L for all n ∈ has an orthogonal basis. Therefore, by Theorem <ref>, the shift σ_ℓ_p(ℝ^2) →ℓ_p(ℝ^2) is conjugate to the product of weighted backward shifts B_ω× B_ω̅, where ω=(ω_n)_n ∈, ω̅=(ω̅_n)_n ∈, ω_n=3+ √(5)/2 and ω̅_n=3- √(5)/2 for any n∈. Therefore,  σ_𝒮 is hyperbolic.




 §.§ Operators without an orthogonal basis


We now address a new class of operators to obtain dynamical information in case the sequence of operators does not have an orthogonal basis.



  §.§.§ Jointly diagonalizable operators




Let X be a finite dimensional Banach space. A sequence 𝒮 = (S_n)_n  ∈ of operators on X is said to be jointly diagonalizable if there is a linear invertible bounded operator ℒ such that, for every n ∈,  there exists a diagonal operator D_n satisfying S_n = ℒ D_n ℒ^-1.



 Let X be a Banach space with dimension 1 ≤ d < +∞. Suppose that the sequence 𝒮 = (S_n)_n  ∈ of operators on X is jointly diagonalizable. Then the shift σ_𝒮ℓ_p(X) →ℓ_p(X) is conjugate to the product of d weighted backward shifts, each one defined on ℓ_p(𝕂). 


We start by proving a general lemma regarding the conjugation of shift operators.


Let X,Y be Banach spaces, = (S_n)_n ∈  be a sequence of linear invertible bounded operators on X and 𝒯 = (T_n)_n ∈ be a sequence of linear invertible bounded operators on Y. Assume that B_X ⊂ X^ℤ and B_Y ⊂ Y^ℤ are Banach spaces such that σ_𝒮 B_X → B_X and σ_𝒯 B_Y → B_Y are well defined shift operators. In addition, suppose that there exists a bounded invertible linear operator H X → Y such that H∘ S_n = T_n ∘ H for every n ∈, and H^ℤ( B_X)= B_Y. Then σ_𝒮 is conjugate to σ_𝒯 by the map ℋ B_X → B_Y, where ℋ((x_n)_n ∈ ℤ) = (H(x_n))_n ∈.


 We note that ℋ is precisely the restriction of H^ to B_X. Therefore, as H is a linear bounded invertible operator and H^ℤ( B_X)= B_Y, so is ℋ B_X → B_Y. Moreover, for any (x_n)_n ∈ ∈ B_X, on has

    ℋ(σ_𝒮((x_n)_n ∈ ))
        =  ℋ((S_n+1(x_n+1))_n ∈ )
     = (H(S_n+1(x_n+1)))_n ∈ 
        =  (T_n+1(H(x_n+1)))_n ∈  = σ_𝒯((H(x_n))_n ∈ )
        =  σ_𝒯(ℋ((x_n)_n ∈ )).




    
Assume that X is a finite dimensional Banach space and let 𝒮 = (S_n)_n  ∈ be a sequence of operators on X jointly diagonalizable by a linear invertible operator ℒ X → X. For every n ∈, take the diagonal operator D_n satisfying S_n = ℒ D_n ℒ^-1. Consider the sequence 𝒟 = (D_n)_n  ∈ of diagonal operators on X and the corresponding shift operator σ_𝒟. Then, by Lemma <ref>, the operators σ_𝒮ℓ_p(X) →ℓ_p(X) and σ_𝒟ℓ_p(X) →ℓ_p(X) are conjugate. Moreover, by Theorem <ref> (cf. Example <ref>), the shift σ_𝒟 is conjugate to a finite product of weighted backward shifts. Hence, so is σ_𝒮. 



[Jointly diagonalizable operators]
Let X= ℝ^2 and consider the sequence = (S_n)_n ∈ of operators in X such that S_n = A for every n ∈, where

    L  =  [ 2 3; 1 2;   ].

The eigenvalues of L are λ = 2 ±√(3) with eigenvectors (±√(3), 1), respectively. As the eigenvectors are not orthogonal and the matrix L is hyperbolic, the image by L of any two orthogonal vectors is not orthogonal. Consequently, 𝒮 = (S_n)_n  ∈ does not have an orthogonal basis. However, as (S_n)_n  ∈ is jointly diagonalizable, by Proposition <ref> we still conclude that σ_𝒮 is conjugate to the product of two weighted backward shift, and is hyperbolic. 





  §.§.§ Anosov family



In Subsubsection <ref> we assumed a joint diagonalization condition on the sequence (S_n)_n  ∈. In what follows, inspired by the shadowing property for non-autonomous Anosov diffeomorphisms which need not admit a common invariant splitting <cit.>, we provide an application of Theorem <ref> where such requirement is relaxed.

[Anosov operators]
Let X = ℝ^2 and consider the hyperbolic, symmetric matrix (hence normal) 

    L =
    [ 2 1; 1 1 ].

Let 0 < λ_-< 1 <λ_+ be the eigenvalues of L with eigenvectors v^- and v^+, respectively. Let ℝ^2 = E^+⊕ E^- be the orthogonal L-invariant splitting given by the eigenspaces E^+ and E^- of L. In particular, there exist convex cones 𝒞^+ and 𝒞^- containing E^+ and E^-, respectively, such that:


  (i) 𝒞^+∩𝒞^-  = {0}

  (ii) L(𝒞^+) ⊊𝒞^+ 

  (iii) L^-1(𝒞^-) ⊊𝒞^-

  (iv) there exists η > 1 such that 

    L(v) > η v  ∀  v∈𝒞^+     and    L^-1(v)  >  η v  ∀  v∈𝒞^-.



In what follows we will choose sequences of matrices whose compositions will preserve the previous cones. Let 𝒰 be a small open neighborhood of L in such a way that, for each T∈𝒰, one has

    T(𝒞^+) ⊊𝒞^+     and     
    T^-1(𝒞^-) ⊊𝒞^-
 
and

    T(v) ≥ η v  ∀  v∈𝒞^+     and    T^-1(v) ≥ η v  ∀  v∈𝒞^-.

Regarding the existence of such an open neighborhood of L we refer the reader to <cit.>.


Let = (S_n)_n  ∈ be a sequence of bounded invertible linear operators on ℝ^2 such that S_0=Id and 

    S_-n∈𝒰  and   
    S_n∈𝒰    ∀ n∈.

This choice, together with the definition of the vectors e_n(·) in (<ref>) and the inclusions  (<ref>), imply that, for any vectors b^+∈^+ and b^-∈^-, one has

    e_n(b^+)  = S_[n+1,0] (b^+)/S_[n+1,0] (b^+)∈ ^+     ∀   n <0

and 

    e_n(b^-)  = ( S_[1,n] )^-1 (b^-)/( S_[1,n] )^-1 (b^-)∈ ^-      ∀   n ⩾ 0.



We proceed by constructing special vectors v_*^+∈^+ and v_*^-∈^- which additionally satisfy 
    e_n(v_*^+)∈ ^+   ∀  n ≥ 0     and     e_n(v_*^-)∈ ^-   ∀ 
    n < 0.
 
Consider the splitting
ℝ^2 = E^+_*⊕ E^-_*, where the subspaces E^+_* and E^-_* are defined by

    E^+_*    =   ⋂_n  ≥  1  S_1… S_n (𝒞^+)   ⊂  𝒞^+ 
    
    E^-_*    =   ⋂_n <  0  
    S_0^-1… S_n+1^-1 (𝒞^-)   ⊂  𝒞^-.

As X=^2, there are unit vectors v_*^±∈ E_*^±∖{0}. If  one considers the basis ={v_*^-,v_*^+} of ℝ^2 then, by construction,  

    S_n+1… S_-1 S_0  (v_*^-) ∈𝒞^-  ∀ n < 0
            and     
    S_n^-1… S_1^-1 (v_*^+)  ∈𝒞^+   ∀ n ≥ 1.

Consequently, e_n(v^±_*) ∈^± for every n∈ℤ.



We claim that the basis  has 𝒮-bounded projections. Indeed, given α, β∈ satisfying |α|+|β| >0, one has 

    α e_n(v_*^-) + β  e_n(v_*^+)^2
        = ⟨α  e_n(v_*^-) + β e_n(v_*^+),  α e_n(v_*^-) + β e_n(v_*^+) ⟩
        = α^2   e_n(v_*^-)^2 
        + 2αβ ⟨ e_n(v_*^-),  e_n(v_*^+)⟩  + β^2  e_n(v_*^+)^2 
        = α^2  + 2αβ ⟨ e_n(v_*^-),  e_n(v_*^+)⟩  + β^2 
        ≥ α^2 + β^2 -  2  |α|  |β| 
        |cosθ| 
        ≥ (1-|cosθ|)  (α^2 + β^2)

where θ∈  ]0,π[ is a (uniform) lower bound for the angle between the vectors e_n(v_*^-) and e_n(v_*^+) for all n ∈ Z, whose existence is ensured by the separation of the cones ^+ and ^-. Consequently, given n ∈, if x = α e_n(v_*^-) + β  e_n(v_*^+) then

    sup_x ∈ ℝ^2 ∖ {0}  Π_e_n(v_*^+)(x)/x    = sup_|α| + |β| > 0 |β|/α e_n(v_*^-) + β  e_n(v_*^+)
        ≤ 1/√(1-|cosθ|)  sup_|α| + |β| > 0  |β|/(α^2 + β^2)^1/2
        ≤ 1/√(1-|cosθ|).
    
Similar estimates are valid for the projections Π_e_n(v_*^-). Therefore,  

    sup_n ∈  sup_b ∈  sup_x ∈ ℝ^2 ∖ {0} Π_e_n(b)(x)/x <  +∞,

proving the claim. 



Now, using Theorem <ref>, we conclude that the shift operator σ_𝒮 is conjugate to the product map B_ω(v_*^-)× B_ω(v_*^+). Furthermore, the weighted backward shifts B_ω(v_*^-) and B_ω(v_*^+) are hyperbolic. In fact, summoning (<ref>), we obtain


    ω_n(v_*^+)  =  S_n+1(e_n+1(v_*^+))/e_n(v_*^+) =   S_n+1(e_n+1(v_*^+)) ≥ η e_n+1(v_*^+) = η >   1     ∀  n ∈ℤ
 
and, similarly, since 

    S_n+1^-1 (S_n+1 (e_n+1(v_*^-))) ≥ η S_n+1 (e_n+1(v_*^-))    ∀  n∈

one gets

    ω_n(v_*^-)  =  S_n+1(e_n+1(v_*^-))/e_n(v_*^-) =   S_n+1(e_n+1(v_*^-)) ≤ η^-1  <   1     ∀  n ∈ℤ.

In particular, by Corollary <ref>, the shift σ_𝒮 satisfies the shadowing property.





  §.§.§ Sequences of elliptic and hyperbolic matrices

In this subsection we will provide examples of shift operators 𝒮 = (S_n)_n ∈ ℤ so that, despite the operators S_n fail to preserve common cones, the shift σ_𝒮 is still conjugate to a finite product of weighted backward shifts. Furthermore, some of these examples satisfy the shadowing property while others do not.   

[Elliptic and hyperbolic matrices]
Consider the linear operators on ℝ^2 given by the matrices

    R_2πζ  = (
    [  cos (2πζ) -sin (2πζ);  sin (2πζ)  cos (2πζ) ]),   ζ∈ℝ∖ℚ    and     
    L  = (
    [   2   0;   0 1/2 ]).

The matrix R_2πζ determines an irrational rotation of the plane, whereas L is a hyperbolic matrix. Thus, matrices R_2πζ's neither preserve any common proper subspaces of ℝ^2 nor admit a common proper invariant cone. In what follows, we will consider sequences 𝒮=(S_n)_n ∈ ℤ of operators on ^2 where S_0=Id and, for each n ∈∖{0}, S_n is either R_2πζ or L, chosen in a way that the shift operator σ_𝒮 is conjugate to a product of weighted backward shifts.



Since L is hyperbolic, there exist convex cones 𝒞^+ and 𝒞^- satisfying the properties (i)-(iv) as in Example <ref>. On the other hand, given  an irrational ζ, there exists a strictly increasing sequence (q_ℓ)_ℓ ∈ of positive integers such that 
lim_ℓ →  +∞  q_ℓ ζ  (mod 1)  =  1. 
Therefore, the sequence of matrices (R_2πζ^q_ℓ)_ℓ ∈ converges, as ℓ goes to +∞, to the identity matrix. Moreover, taking a subsequence if necessary, we may assume without loss of generality that, for every ℓ∈, 

    R_2πζ^q_ℓ (  L(^+) ) ⊊ ^+     and     
    L(R_2πζ^q_ℓ ( ^+ )) ⊊ ^+


    L^-1(R_2πζ^-q_ℓ ( ^- )) ⊊ ^-     and     R_2πζ^-q_ℓ (  L^-1(^- ))) ⊊ ^-.




We will now consider sequences 𝒮=(S_n)_n ∈ ℤ of the form

    …L … L_n_2 R_2πζ… R_2πζ_m_1 L … L_n_1   ↓Id L … L_n_1 R_2πζ… R_2πζ_m_1 L … L_n_2…

for suitable choices of sequences of positive integers (n_i)_i ∈ and (m_i)_i ∈, where ↓ marks the position zero. Given any such sequence 𝒮 of linear operators,
consider the splitting
ℝ^2 = E^+_*⊕ E^-_*, where the subspaces E^+_* and E^-_* are defined by

    E^+_*    =   ⋂_ℓ ≥ 1  L^n_1 R_2πζ^m_1… L^n_ℓ R_2πζ^m_ℓ  (𝒞^+)  ⊂ 𝒞^+ 
    
    E^-_*    =   ⋂_ℓ ≥  1  
     L^-n_1 R_2πζ^-m_1… L^-n_ℓ R_2πζ^-m_ℓ  (𝒞^-)  ⊂ 𝒞^-.

If v_*^±∈ E_*^±∖{0} are unit vectors and one considers the basis 
= {v_*^-,v_*^+} of ℝ^2 then, by construction, as the cones 𝒞^- and 𝒞^+ are sufficiently separated and invariant (cf. (<ref>) and (<ref>)), there exists θ>0 such that 

    ∡ (e_n(v_*^+), e_n(v_*^-)) ≥ θ > 0     ∀ n∈.

In particular, due to Corollary <ref>,  has 𝒮-bounded projections.
Therefore, by Theorem <ref>,
the shift operator σ_𝒮 is conjugate to the product B_ω(v_*^+)× B_ω(v_*^-). 




[Selection of the frequencies (n_i)_i ∈ and (m_i)_i ∈]
We may change the frequency with which L occurs in the sequence (S_n)_n ∈ ℤ, as illustrated in (<ref>), to study its impact on the properties of the shift operator σ_𝒮. We consider the following two cases.



(1) Unbounded gaps: sup_i ∈  m_i  =  +∞.


In this case, for every k∈,

    lim sup_n → +∞  S_-k-n …  
    S_-k(v_*^-)^1/n >  1     and    lim inf_n → +∞  S_k+n^-1 … S_k^-1 (v_*^+)^-1/n <  1.

 However, since there exist sufficiently large numbers of consecutive occurrences of rotation matrices R_2πζ in the sequence 𝒮=(S_n)_n ∈, we conclude that, for any n∈ there is k∈ such that, for all x ∈^2 ∖{0},

    S^-1_k+n… S_k^-1 (x) = S_-k-n… S_-k (x)  =  x.

This way, the sequence 𝒮 does not satisfy  
condition (C) in Corollary <ref>.
The presence of arbitrary large numbers of consecutive occurrences of the elliptic matrices R_2πζ also prevent conditions (A) and (B) in 
Corollary <ref> to hold.
Therefore, σ_𝒮 does not satisfy the shadowing property.


(2)  Bounded gaps: sup_i ∈  m_i  <  +∞.



Let M = 1+sup_i ∈  m_i. We note that 

    inf_k  ∈   # {k ≤ j ≤ k+M  S_j = L }/M ≥ 1/M.

Using (<ref>), we know that, for any n∈ℤ,

    ω_n(v_*^+)  =  S_n+1 (e_n+1(v_*^+))  and  ω_n(v_*^-)  =  S_n+1 (e_n+1(v_*^-))

and so, whenever S_n+1 = L, one has

    ω_n(v_*^+)  ≥ η  >   1   and  ω_n(v_*^-)  ≤ η^-1 <   1     ∀  n ∈;

otherwise, S_n+1 = R_2πζ and

    ω_n(v_*^+)  = ω_n(v_*^-) = 1.
 
Therefore, for every k ∈ and n ∈,  

    |ω_k(v_*^+) …ω_k+n(v_*^+)|^1/n  =  η^  # {k  ≤ j  ≤ k+n   S_j = L }/n

which yields

    inf_k ∈  |ω_k(v_*^+) …ω_k+n(v_*^+)|^1/n =inf_k ∈  η^ # {k ≤  j  ≤  k+n  S_j = L }/n ≥ η^ inf_k ∈  # {k ≤  j  ≤  k+n  S_j = L }/n.

Summoning (<ref>), we conclude that for the positive integer M one has

    inf_k ∈  |ω_k(v_*^+) …ω_k+M(v_*^+)|^1/M  ≥  η^ inf_k ∈  # {k ≤  j  ≤  k+M  S_j = L }/M ≥  η^1/M >  1.

Thus, by <cit.>), the following limit exists and satisfies 

    lim_n → +∞  inf_k ∈  |ω_k(v_*^+) …ω_k+n(v_*^+)|^1/n >  1.

Analogous estimates show that

    lim_n → +∞  sup_k ∈  |ω_k(v_*^-) …ω_k+n(v_*^-)|^1/n <  1.

Therefore, 
item (C) of Corollary <ref> implies that the shift operator σ_𝒮 satisfies the shadowing property. Moreover, as σ_𝒮 is conjugate to the product of two weighted backward shifts with the shadowing property, we have that it is generalized hyperbolic.








  §.§.§ Shift operators conjugate to skew-products of weighted shifts

We now present an example of a sequence 𝒮=(S_n)_n ∈ ℤ without 𝒮-bounded projections and whose shift operator is not linearly conjugate to the product of the two weighted backward shifts associated to the weights ω((1,0)) and ω((0,1)); it turns out to be conjugate to a skew-product of weighted backward shifts.


[𝒮-unbounded projections] 

Let X=ℝ^2 endowed with the norm given by (x,y)=max {|x|,|y|} and 𝒮=(S_n)_n ∈ be the sequence of bounded linear operators expressed by the matrix 

    S_n  =  L  =  [ 1 1; 0 1 ]    ∀  n∈.

A simple computation shows that, for every (x_n,y_n)_n ∈ ∈ℓ_p(ℝ^2) and any k∈, one has

    σ_𝒮^k ( (x_n,y_n)_n ∈  ))  
     =  (x_n+k+k  y_n+k,  y_n+k)_n ∈ .

Moreover, for every n ∈, 

    e_-n(1,0)=L^n(1,0)/L^n(1,0)=(1,0)
            and     
    e_-n(0,1)=L^n(0,1)/L^n(0,1)=(n,1)/n=(1, 1/n).

Hence, the angle between these two vectors goes to zero as n→+∞ and

    Π_e_-n(0,1) ≥ Π_e_-n(0,1)((0,1))/(0,1) = Π_e_-n(0,1)(-n e_-n(1,0) + n  e_-n(0,1)) =   n.

Consequently, the canonical basis ={(1,0),(0,1)} does not have 𝒮-bounded projections. A similar argument shows that every basis  of ℝ^2 has no 𝒮-bounded projections.  







We now verify that the shift operator σ_𝒮ℓ_p(ℝ^2) →ℓ_p(ℝ^2) is conjugate to the skew-product of two weighted backward shifts. By the choice of the norm in ℝ^2, the linear map ϕℓ_p(ℝ^2) →ℓ_p(ℝ) ×ℓ_p(ℝ) given by 

    ϕ((x_n,y_n)_n ∈ ) = ((x_n)_n ∈ ,  (y_n)_n ∈ )

is an isomorphism.

Moreover,

    ϕ∘σ_𝒮∘ϕ^-1 ((x_n)_n ∈ ,   (y_n)_n ∈ ) 
        =  (ϕ∘σ_𝒮)  ((x_n,y_n)_n ∈ ) 
        = ϕ ((x_n+1 + y_n+1,  y_n+1)_n ∈ ) 
        = ((x_n+1 + y_n+1)_n ∈ , (y_n+1)_n ∈ ) 
        = ((x_n+1)_n ∈  + (y_n+1)_n ∈ ,  (y_n+1)_n ∈ ) 
        = σ_𝒮 ((x_n)_n ∈ ,  ,(y_n)_n ∈ )

where σ_𝒮 is the skew-product 
defined by 

    [                                                      σ_𝒮                                           ℓ_p(ℝ) ×ℓ_p(ℝ)                                                        →                                           ℓ_p(ℝ) ×ℓ_p(ℝ);                                                                                         ((x_n)_n ∈ ,  (y_n)_n ∈ )                                                        ↦ (B_ω ((x_n)_n ∈ ) + B_ω ((y_n)_n ∈ ),  B_ω ((y_n)_n ∈ )) ]

and B_ω is the weighted shift given by

    B_ωℓ_p(ℝ)→ℓ_p(ℝ)

with weights ω = (ω_n)_n  ∈ and   ω_n = 1 for all n∈.



This example shows that Theorem <ref> is sharp, that is, despite the fact that ℰ = {(1,0),(0,1)} is a basis for ^2, σ_𝒮 fails to be linearly conjugate to B_ω((1,0))× B_ω((0,1)). Indeed, an easy computation shows that the weights for B_ω((1,0)) are

    ω_n = 1     ∀ n ∈
 
and the weights for B_ω((0,1)) are

    ω_n = 
        1     n=0,1
    n-1/n    n ≥ 2
    |n|+1/|n|    n ≤ -1.

As the weights of B_ω((0,1)) are bi-Lipschitz equivalent to the constant sequence 1, we have that B_ω((0,1)) is linearly conjugate to B_ω((1,0)). In particular, if we let T = B_ω((1,0))× B_ω((0,1)), then all orbits of T are bounded. However, if v  =  (0,y_n)_n ∈ ∈ℓ_p(^2) with some y_n_0≠ 0, from (<ref>) we deduce that

    lim_k  →  +∞ σ_𝒮^k(v) =   +∞,
 
implying that σ_S is not linearly conjugate to B_ω((1,0))× B_ω((0,1)).


We note that the shift operator σ_𝒮 does not have the shadowing property and hence it is not generalized hyperbolic. 
This is so because the shadowing property is preserved by factors, and B_ω((1,0)), a factor of σ_𝒮, does not have the shadowing property as all of its weights are 1's. 





 



§ PROOFS




 §.§ Dissipative operators on Banach spaces are shift operators


This subsection is devoted to the proof of Theorem <ref>. Let (E_n)_n ∈ ℤ be a dissipative decomposition of X, where  E_0⊂ X is a closed subspace, E_n = T^n(E_0) for every n∈ℤ and

    X = ⊕_n ∈ ℤ E_n.


Given x∈ X, there exist unique vectors x_n∈ E_n, for n∈ℤ, such that x = ∑_n ∈ ℤ x_n. Let H X →E_0^ℤ be defined by

    H(x) = (T^n(x_-n))_n ∈ ℤ.
 
By the uniqueness of the decomposition and the linearity of T, the map H is well defined, linear and injective. We endow the vector space B = H(X) with the norm ·_ B given, for each (y_n)_n ∈ ℤ∈ B, by

    (y_n)_n ∈ ℤ_ B = x_X

where H(x) = (y_n)_n ∈ ℤ. Therefore, H is continuous and B is a Banach space. Thus, by the Open mapping theorem, H is open.



Now consider the sequence of linear operators 𝒮=(S_n)_n ∈ ℤ acting on E_0 so that S_n is the identity for each n∈ℤ. We claim that 

    σ_𝒮∘ H = H∘ T

where 
σ_𝒮 B→ B is the shift operator determined by 𝒮. Indeed, let x∈ X given by x=∑_n ∈ ℤ x_n, with x_n∈ E_n. Then T(x) = ∑_n ∈ ℤ T(x_n), and so 

    H(T(x))  = (T^n(T(x_-n-1)))_n ∈ ℤ = (T^n+1(x_-(n+1)))_n ∈ ℤ.

Analogously, 

    σ_𝒮(H(x))
         =  σ_𝒮((T^n(x_-n)_n ∈ ℤ)
         = (T^n+1(x_-(n+1)))_n ∈ ℤ.

This completes the proof of the theorem.




 §.§ Dissipative composition operators are shift operators



In this subsection we will prove Theorem <ref>.
Assume (M,𝔅,μ, f)  and T_f are as stated in Theorem <ref>. We need to find a suitable Banach space Y, a sequence 𝒮= (S_n)_n  ∈ of invertible linear operators  on Y and a Banach space B⊂ Y^ℤ, such that ( B, σ_ 𝒮) is conjugate to T_f.


To this end, let Y = L^p (W, μ) and consider the vector space  

    B = { (ψ_n)_n ∈ ℤ∈ Y^ℤ∑_n ∈ ℤ ∫_W |ψ_n|^p  dμ f^n/dμ  dμ < +∞}
 
endowed with the norm 

    (ψ_n)_n ∈ ℤ_ B = (∑_n ∈ ℤ ∫_W |ψ_n|^p  dμ f^n/dμ  dμ)^1/p.

Next we define a map Γ L^p(M, μ) → B as follows: for each φ∈ L^p(M, μ), let ψ = Γ (φ) ∈ B be given by ψ_n = φ∘ f^n. Note that the domain of ψ_n is W. 


We begin by showing that Γ is well defined and 1-Lipschitz: 

    Γ (φ) _ B ^p 
         = 
         ψ_ B^p
             =  ∑ _n  ∈  ∫_W |ψ_n|^p  dμ f^n/dμ dμ
        = ∑ _n  ∈  ∫ _W | ψ_n|^p  dμ f^n
        = ∑ _n  ∈  ∫ _f^n(W) |ψ_n|^p∘ f^-n  dμ
        = ∑ _n  ∈  ∫ _f^n(W) |φ|^p   dμ
        =  φ_L^p(M, μ).

Afterwards, we show that Γ maps onto B and its inverse is also 1-Lipschitz. Indeed, let ψ  = (ψ_n)_n  ∈ ∈ B. We define  φ M → M by φ (x) = ψ_n(f^-nx), where n ∈ is the unique integer such that x ∈ f^n(W). Then
    
    φ_L^p(M)^p     = ∑ _n  ∈ ∫ _f^n(W) | φ|^p   dμ
        =∑ _n  ∈ ∫ _f^n(W) | ψ_n |^pf^-n dμ
        = ∑ _n  ∈ ∫ _W | ψ_n|^p  d μ f^n
        = ∑ _n  ∈ ∫ _W | ψ_n|^p  d μ f^n/d μ  dμ
       =  ψ_ B^p

confirming that Γ is surjective and its inverse is 1-Lipschitz as well. Consequently, Γ is an isometry and B is a Banach space. 


For each n ∈, let  S_n Y → Y be the identity map and = (S_n)_n  ∈. We next show that Γ is a conjugacy between T_f and σ_.  Indeed, given φ∈ L^p(M, μ), one has

    Γ (T_f(φ))   = Γ (φ∘ f) 
      = ((φ∘ f)f^n|_W)_n ∈  Z = (φ∘ f^n+1|_W)_n ∈

and 

    σ_(Γ (φ))   = σ_((φ∘ f^n|_W)_n  ∈ )   =  (φ∘ f^n+1|_W)_n  ∈ Z.

The proof of the theorem is complete.


Whenever there is C >0 such that 

    1/C < dμ f^n/d μ_∞ <  C     ∀  n ∈

the Banach space B provided by Theorem <ref> is isomorphic to ℓ_p( L^p(μ|_W)) and σ_𝒮 is conjugate to the usual shift map on ℓ_p( L^p(μ|_W)). 







 §.§ Weighted shifts as factors of shift operators


Here we prove Theorem <ref>. Let ℰ be an orthogonal basis for 𝒮=(S_n)_n∈ Z.
Given x∈ℰ recall that 

    e_n(x)  = [ ( S_[1,n] )^-1 (x)/( S_[1,n] )^-1 (x)                            if   n ≥ 0;           S_[n+1,0] (x)/S_[n+1,0] (x)                             if   n <0 ].

and that, by construction of the weights, one has 

    S_n+1 (e_n+1(x)) = ω_n+1(x)  e_n(x)      ∀  n ∈ Z.

Consider the function 
Γ=Γ_xℓ_p(X) →ℓ_p(𝕂) given by 

    Γ ((x_n)_n ∈ ℤ) = (Π_e_n(x) (x_n))_n  ∈

where Π_e_n(x) X → F_n(x) stands for the orthogonal projection from X onto the subspace F_n(x) generated by the unit vector e_n(x). We claim that the linear map Γℓ_p(X) →ℓ_p(𝕂) is continuous and surjective. 

As X is a Hilbert space, all orthogonal projections Π_z are uniformly bounded (with norm one), and so 

    Γ((x_n)_n ∈ )_p^p 
         = ∑_n ∈  Π_e_n(x) (x_n)^p
         ≤ ∑_n ∈  x_n^p
         = (x_n)_n ∈ )_p^p.

This proves the continuity of Γ and also that Γ((x_n)_n ∈ ) belongs to ℓ_p(𝕂).

To show the surjectivity of Γ we just observe that, given (t_n)_n ∈ ℤ∈ℓ_p( 𝕂), if we consider u_n = t_n e_n(x) for every n∈, then 

    Γ((u_n)_n ∈ ℤ)  = Γ((t_n e_n(x))_n ∈ ℤ)  = (t_n Π_e_n(x) (e_n(x)))_n ∈ ℤ = (t_n)_n  ∈ .

Moreover, since all vectors e_n are unitary, it is clear that (u_n)_n ∈ ℤ_p = (t_n)_n ∈ ℤ_p.


We are left to prove that Γ∘σ_𝒮=B_ω(x)∘Γ. On the one hand, given (x_n)_n ∈ ℤ∈ℓ_p( X), 

    Γ (σ_𝒮 ((x_n)_n ∈ ℤ))    =   Γ ((S_n+1(x_n+1))_n ∈ ℤ) 
       =   (Π_e_n(x)(S_n+1(x_n+1)))_n ∈ ℤ
       =   (Π_1/ω_n+1 S_n+1 (e_n+1(x))(S_n+1(x_n+1)))_n ∈ ℤ

and

    B_ω(x) (Γ ((x_n)_n ∈ ℤ))    =    B_ω(x) ((Π_e_n(x) (x_n))_n ∈ ℤ) 
       =   (ω_n+1 Π_e_n+1(x) (x_n+1))_n ∈ ℤ.

On the other hand, we may write x_n+1 as a sum

    x_n+1 =  λ_n+1  e_n+1(x) + γ_n+1

where λ_n+1∈ and the vector γ_n+1∈ X is orthogonal to e_n+1(x). Moreover, for each b∈∖{x}, there exists η_b∈𝕂 such that 

    γ_n+1  = ∑_b ∈ ,  b ≠  x η_b  e_n+1(b).

Therefore,

    S_n+1 (x_n+1)      =  λ_n+1  S_n+1  (e_n+1(x)) + 
    ∑_b ∈ ,  b ≠  x η_b  S_n+1  (e_n+1(b))

where 

    {S_n+1(e_n+1(b))  b ∈} =  {ω_n+1(b) e_n(b)  b ∈}

is a set of pairwise orthogonal vectors by the assumption on the sequence 𝒮. Thus,

    Π_1/ω_n+1(x) S_n+1 (e_n+1(x))(S_n+1 (x_n+1))   =  ω_n+1(x) λ_n+1 = ω_n+1(x)  Π_e_n+1(x) (x_n+1).

This ends the proof of the theorem.







 §.§ Classification theorems




  §.§.§ Proof of Theorem <ref>


The statement of the theorem is immediate if the Hilbert space X has  dimension d = 1, since in this case the map σ_𝒮 is, by definition, a weighted backward shift whose weights are precisely (ω_n)_n  ∈.


Let X be a Hilbert space with dimension d ≥ 2 and = {b_1, …, b_d} be an orthogonal basis for (S_n)_n ∈, that is, _n = {e_n(x)  x∈} is an orthonormal basis on X for every n ∈. Endow the vector space (ℓ_p(𝕂))^d with the sum norm. Consider the map

    Iℓ_p(X)   →  (ℓ_p(𝕂))^d

given by

    I((x_n)_n ∈ )
       =   ( Π_e_n(b_1)(x_n),
    Π_e_n(b_2)(x_n),
    …, 
    Π_e_n(b_d)(x_n) )_n ∈ 
       =   ( Γ_b_1((x_n)_n  ∈ ),
    Γ_b_2((x_n)_n  ∈ ),
    …, 
    Γ_b_d((x_n)_n  ∈ ) )_n ∈ .

It is clear that I is linear and injective. 

Moreover, I is surjective and continuous. Indeed, given 

    ((α_n,1)_n ∈ , (α_n,2)_n ∈ , …, (α_n,d)_n ∈)   ∈   (ℓ_p(𝕂))^d

let 

    x_n  = ∑_i=1^d  α_n,i  e_n(b_i)   ∈ X

for all n∈. By definition, 

    I((x_n)_n ∈ )=((α_n,1)_n ∈ ,(α_n,2)_n ∈ , …, (α_n,d)_n ∈ ).


Moreover, (x_n)_n  ∈ belongs to ℓ_p(X) since

    x_n^p
            =  ∑_i=1^d  α_n,i  e_n(b_i) _X^p 
        ≤ (∑_i=1^d  α_n,i  e_n(b_i) _X)^p 
        ≤  d^p-1∑_i=1^d  α_n,i  e_n(b_i) _X^p 
        =   d^p-1∑_i=1^d  
        |α_n,i|^p

where the third inequality is due to the fact that, as p ≥ 1, the map t ≥ 0 ↦ t^p is convex, and in the last equality we use the assumption that _n is a normalized basis. Therefore

    ∑_n ∈  x_n^p
            ≤ ∑_n ∈  d^p-1 ∑_i=1^d   |α_n,i|^p 
        =   d^p-1 ∑_i=1^d   ∑_n ∈      |α_n,i|^p  < +∞.

By the Open mapping theorem, I is an isomorphism. Furthermore, by construction,

    (I∘σ_𝒮)((x_n)_n ∈ )
            = I((S_n+1(x_n+1))_n ∈ ) 
        = (Γ_b_1((S_n+1(x_n+1))_n ∈  Z), …,  Γ_b_d((S_n+1(x_n+1))_n ∈  Z) )
        
       
        =   ( B_ω(b_1)(Γ_b_1((x_n)_n ∈ )), …,  
        B_ω(b_d)(Γ_b_d((x_n)_n ∈ ) ))
        
        = (B_ω(b_1)× B_ω(b_2)×…× B_ω(b_d))
        (I((x_n)_n ∈  Z)).

This completes the proof of the theorem.




  §.§.§ Proof of Theorem <ref>

Fix 1≤ p < +∞ and let = {b_1, …, b_d} be a basis of X with 𝒮-bounded projections. For each n∈, consider the basis _n={e_n(x)  x∈}. Given b∈, take the linear map 

    [                     Γ_b                  ℓ_p(X)                       →                     𝕂^ℤ;                                     (x_n)_n ∈ ℤ                       ↦ (Π_e_n(b) (x_n))_n  ∈ . ]

We claim that Γ_b is continuous and Γ_b(ℓ_p(X))=ℓ_p(𝕂). Indeed, by the 𝒮-bounded projections assumption (<ref>), there exists C>0 such that 

    Γ_b((x_n)_n ∈ )_p^p 
         = ∑_n ∈  Π_e_n(b) (x_n)^p
         ≤  C^p ∑_n ∈  x_n^p
         =  C^p (x_n)_n ∈ )_p^p.

This estimate proves the continuity of Γ_b and also that Γ_b((x_n)_n ∈ ) ∈ℓ_p(). Conversely, given (t_n)_n ∈ ℤ∈ℓ_p(𝕂), the vector (t_n e_n(b))_n ∈ ℤ belongs to ℓ_p(𝕂) because the elements in _n are normalized, and one has 

    Γ_b((t_n e_n(b))_n ∈ ℤ)   =  (t_n)_n ∈ ℤ.


In order to construct the claimed conjugation we will need the following auxiliary result.



 Let X be a finite dimensional Banach space, 
𝒮=(S_n)_n ∈ ℤ be a sequence of operators on X and  be a basis of X with 𝒮-bounded projections. There exists 𝒦_p>0 such that, for every (α_b)_b ∈ ∈𝕂^ X and n∈, one has

    ∑_b ∈  α_b   e_n(b) ^p   ≤ 𝒦_p ∑_b ∈  α_b   e_n(b) ^p  = 𝒦_p  ∑_b ∈    |α_b|^p.



 Consider 1 < p < +∞ and take q ∈ such that 1/p + 1/q = 1. Given x ∈ X, write 

    x  = ∑_b ∈  α_b   e_n(b)   =  ∑_b ∈  Π_e_n(b) (x).
    
Then, using the triangular inequality, the assumption (<ref>) and Hölder inequality, we get

    x^p 
         =  ∑_b ∈  Π_e_n(b)(x)^p 
     ≤ (∑_b ∈  C  α_b  e_n(b))^p  
        =  C^p (∑_b ∈   |α_b|· 1 )^p 
        ≤  C^p  [(∑_b ∈   |α_b|^p)^1/p (∑_b ∈   1^q)^1/q]^p 
        ≤  C^p  ( X)^p/q ∑_b ∈   |α_b|^p 
        =  C^p ( X)^p (p-1) ∑_b ∈   |α_b|^p.


In case p=1, given x ∈ X written as

    x  = ∑_b ∈  α_b   e_n(b)   =  ∑_b ∈  Π_e_n(b) (x)
 
then, using the triangular inequality and the assumption (<ref>), we get

    x  =  ∑_b ∈  Π_e_n(b)(x) ≤ ∑_b ∈  C  α_b  e_n(b) =  C ∑_b ∈   |α_b|.




Consider the map Iℓ_p(X) → (ℓ_p(𝕂))^d defined by

    I((x_n)_n ∈ )
       =   ((Π_e_n(b_1)(x_n))_n ∈ ,  (Π_e_n(b_2)(x_n))_n ∈ , …,  (Π_e_n(b_d)(x_n))_n ∈  )
       =   (Γ_b_1((x_n)_n  ∈ ), Γ_b_2((x_n)_n  ∈ ), …,  Γ_b_d((x_n)_n  ∈ )
    )_n ∈

By construction, I is linear, injective and continuous. Moreover, I is surjective since, given 

    ((α_n,1)_n ∈ ,(α_n,2)_n ∈ , …, (α_n,d)_n ∈ )  ∈  (ℓ_p(𝕂))^d

one has 

    I((∑_i=1^d α_n,i e_n(b_i))_n ∈ )  = ((α_n,1)_n ∈ ,(α_n,2)_n ∈ , …, (α_n,d)_n ∈ )

and, by using Lemma <ref>, 

    ∑_n ∈  ∑_i=1^d  α_n,i e_n(b_i) ^p  ≤ ∑_n ∈  𝒦_p ∑_i=1^d   |α_n,i|^p 
      = 𝒦_p ∑_i=1^d   ∑_n ∈  
        |α_n,i|^p   < +∞.

Therefore, I is an isomorphism. Furthermore, by construction,

    (I∘σ_𝒮)((x_n)_n ∈ )
            = I((S_n+1(x_n+1))_n ∈ ) 
        = (Γ_b_1((S_n+1(x_n+1))_n ∈  Z), …,  Γ_b_d((S_n+1(x_n+1))_n ∈  Z) )
        
       
        =   ( B_ω(b_1)(Γ_b_1((x_n)_n ∈ )), …,  
        B_ω(b_d)(Γ_b_d((x_n)_n ∈ ) ))
        
        = (B_ω(b_1)× B_ω(b_2)×…× B_ω(b_d))
        (I((x_n)_n ∈  Z)).

This finishes the proof of the theorem.



  §.§.§ Proof of Corollary <ref>


Let X be a finite dimensional Hilbert space with dimension d ≥ 2 and assume that the sequence 𝒮=(S_n)_n ∈ has a normalized basis = {v_1, ⋯, v_d} of X for which there exists 0 < γ < 1/(d-1) such that, for every n ∈, the angle ∡(u, v) between distinct vectors u,v ∈_n satisfies the condition

    cos∡(u, v)  ∈  [-γ, γ].
 

Consider x ∈ X ∖{0} whose coordinates with respect to the basis _n are

    x  =  α_n,1 e_n(v_1) + ⋯ + α_n,d e_n(v_d)

where α_n,i∈ for every 1 ≤ i ≤ d. Then 

    x_X^2    =   α_n,1  e_n(v_1) + ⋯ + α_n,d e_n(v_d)_X^2  
       =   α_n,1^2 + ⋯ + α_n,d^2   + ∑_1  ≤  i,    j ≤ d α_n,i  α_n,j <e_n(v_i), e_n(v_j)> 
       =   α_n,1^2 + ⋯ + α_n,d^2   + ∑_1  ≤  i <    j ≤ d 2 ℜ𝔢(α_n,i  α_n,j)  cosθ_n,i,j
       ≥   α_n,1^2 + ⋯ + α_n,d^2   - ∑_1  ≤  i <    j ≤ d 2  |ℜ𝔢(α_n,i  α_n,j)|  |cosθ_n,i,j| 
       ≥   α_n,1^2 + ⋯ + α_n,d^2   - ∑_1  ≤  i <    j ≤ d 2 α_n,i  α_n,j |cosθ_n,i,j| 
       =   α_n,1^2 + ⋯ + α_n,d^2   - ∑_1  ≤  i <    j ≤ d 2 α_n,i α_n,j |cosθ_n,i,j| 
       ≥   α_n,1^2 + ⋯ + α_n,d^2   - ∑_1  ≤  i <    j ≤ d  2 α_n,i α_n,j γ
       ≥   α_n,1^2 + ⋯ + α_n,d^2   - ∑_1  ≤  i  <  j ≤ d (α_n,i^2 + α_n,j^2) γ
       ≥   ∑_i=1^d  α_n,i^2  (1 - ∑_j ≠ iγ) 
       =   (α_n,1^2 + ⋯ + α_n,d^2)  (1 - (d-1) γ).



Observe now that, by the choice of γ, one has 1 - (d-1) γ > 0. Consequently, for every n ∈ and j ∈{1,⋯, d}, 

    sup_x  ≠ 0 Π_e_n(v_j)(x)_X/x_X   =   α_n,j/x_X
       ≤   α_n,j/√((1 - (d-1) γ)   (α_n,1^2 + ⋯ + α_n,d^2))
       ≤   1/√((1 - (d-1) γ)).

That is, the basis  has 𝒮-bounded projections. Therefore, by Theorem <ref>, for every 1 ≤ p < +∞ the shift operator σ_𝒮ℓ_p(X) →ℓ_p(X) is conjugate to the product of weighted backward shifts ∏ _x  ∈  B_ω(x)(ℓ_p(𝕂))^ X→(ℓ_p(𝕂))^ X. The proof of the corollary is complete.



We note that, when d=2, the demand on the angles 
{θ_n,i,j n ∈,  1 ≤ i ≠ j ≤ 2} in the previous proof reduces to requesting that they do not approach neither 0 nor π. The next example shows that, when d > 2, this is no longer enough to guarantee that the basis  has 𝒮-bounded projections and we need a sharper uniform bound. 

[Unbounded projections]
Assume that d = 3 and 𝕂 = ℝ. Given a suf­ficiently small δ >0, we may choose a normalized basis _δ = {e_1, e_2, e_3} of ^3 with angles θ_i,j sufficiently close to 2π/3 so that cos (θ_1,2) = -1/2, 
cos  (θ_1,3) = cos  (θ_2,3) =-1/2 + δ. If  x = α_1 e_1 + α_2  e_2 + α_3   e_3, then 

    x^2    =   α_1  e_1 + α_2  e_2 + α_3  e_3^2  
       =   α_1^2 + α_2^2 + α_3^2  + 2 α_1 α_2 cos (θ_1,2)  + 2 α_1 α_3 cos (θ_1,3) + 2 α_2 α_3 cos (θ_2,3) 
       =   α_1 (α_1 + 2α_2 cos (θ_1,2) )  + α_2 (α_2  +   2 α_3 cos (θ_2,3) )   +  α_3 (α_3    + 2  α_1cos (θ_1,3)).

In the particular case of α_1 =α_2 = α_3 = 1 and x_0 = e_1 + e_2 + e_3, we get

    x_0^2  =  0 + 1 (2 δ) + 1 (2δ)    = 4 δ

and so, for 1 ≤ j ≤ 3,

    sup_x  ≠ 0 Π_e_j(x)/x ≥ Π_e_j(x_0)/x_0  =  |α_j|/x_0 = 1/4 δ.

As δ may be chosen arbitrarily small, the projection Π_e_j is not uniformly bounded independent of δ. We note that, as δ goes to 0, the space generated by {e_1, e_2, e_3} becomes 2-dimensional, so the angle between e_3 and the plane generated by 
{e_1, e_2} approaches 0. 






  §.§.§ Proof of Corollary <ref>


The next proposition provides another sufficient condition for a sequence 𝒮 of bounded linear invertible operators to have 𝒮-bounded projections. Corollary <ref> is an immediate consequence of this proposition and  Theorem <ref>.


Let X be a finite dimensional Hilbert space, ={v_1,v_2, …, v_d} be a basis of unit vectors of X and 𝒮 = (S_n)_n ∈ be a sequence of bounded linear invertible operators on X. For each 1≤ i ≤ d, consider the vector space F_n,i⊂ X generated by the vectors {e_n(v_j)  1≤ j ≤ d,   j≠ i}. If


    inf_n ∈ ℤ,    1 ≤  i ≤  d  ∡ (e_n(v_i),F_n,i)  >  0

then the basis  has 𝒮-bounded projections.



From the assumption (<ref>), we deduce that there exists θ∈  ]0, π[ such that, for every n ∈ℤ and 1 ≤ i ≤ d, one has

    |cos(∡ (e_n(v_i),F_n,i))| ≤ |cos θ|   <   1.

Fix n ∈ and 1 ≤ i ≤ d. For each x∈ X, write x = α_i   e_n(v_i) + f_n,i, where f_n,i∈ F_n^i is unique. In what follows, assume that x ≠ 0. 
Using the estimate

    x^2       ≥ α_i^2  e_n(v_i)^2 + 
        f_n,i^2 - 2e_n(v_i) f_n,i |cosθ| 
        =  α_i^2 + f_n,i^2 - 2f_n,i  |cosθ|
         >  0

we obtain

    Π_e_n(v_i)(x)^2/x^2    ≤ α_i^2/α_i^2 + 
        f_n,i^2 - 2 α_i f_n,i  |cosθ|
       
        ≤ α_i^2/(α_i - f_n,i)^2 
        + (2 - 2 |cosθ|) α_i f_n,i.

If α_i=0, the previous quotient is zero and there is nothing to prove. If α_i>0, writing

    t   =   f_n,i/α_i
    β   =     1 - |cosθ|   >  0

then (<ref>) becomes 

    Π_e_n(v_i)(x)^2/x^2    ≤ α_i^2/(1-t)^2  α_i^2 
        + 2 β t α_i^2
        ≤ 1/(1-t)^2 + 2 β t
        ≤ 1/min_s ≥ 0 {(s+β-1)^2 + 2 β-β^2}
         = 1/2 β-β^2.

Thus, Π_e_n(v_i) is uniformly bounded, independently on n and i. This proves the proposition.






Example <ref> shows that, without the assumption (<ref>), the conclusion of Proposition <ref> may fail. 





 §.§ Proof of Corollary <ref>


Let X be a d-dimensional Hilbert space and  be an orthogonal basis for (S_n)_n  ∈. In what follows, let property 𝒫 stand for transitivity, mixing, or shadowing. Under the assumptions of Corollary <ref>, Theorem <ref> ensures that the shift operator σ_𝒮 is conjugate to the product of weighted backward shifts ∏ _x  ∈  B_ω(x) (ℓ_p(𝕂))^d → (ℓ_p(𝕂))^d. 
Hence, Corollary <ref> is a direct consequence of the next lemma.


Let Y be a Banach space and T_i  Y → Y, 1≤ i ≤ d, be a sequence of invertible linear operators. Consider T=T_1× T_2 
×…× T_d  Y^d→ Y^d, where Y^d is endowed with the sum norm. Then

    T  has property 𝒫  ⇒  {T_i 1≤ i ≤ d}  has equi-property 𝒫.




Assume that 𝒫 is transitivity or mixing. Given any non-empty open sets U,V⊂ Y, it suffices to apply the corresponding property for the operator T and the open subsets U× U ×…× U and V× V ×…× V of Y^d to obtain the equi-property 𝒫 for the class of operators {T_i 1≤ i ≤ d} acting on Y.


Suppose now that 𝒫 is the shadowing property. By definition, there is K >0 which witnesses the shadowing property of T. We claim that the same K verifies equi-shadowing for {T_i 1≤ i ≤ d}. Indeed, for 1 ≤ i ≤ d, for any bounded sequence (z^i_n)_n  ∈ in Y, there is a bounded sequence (x_n)_n  ∈ in Y^d satisfying 

    x_n+1-T(x_n)  =  (0, …, 0, z^i_n, 0, …, 0 )    and    sup_n ∈  x_n < K sup_n  ∈ z_n^i.
 
Letting x^i_n be the ith coordinate of x_n, we get

    x^i_n+1-T_i(x^i_n)  =   z^i_n,     and    sup_n ∈  x^i_n < K sup_n  ∈  z_n^i

completing the proof of the lemma.







§ OPEN QUESTIONS

We showed in Corollary <ref> that, under an extra assumption, the answer to the following questions is yes. However, the general problem remains open.

For a general shift operator, is the shadowing property equivalent to generalized hyperbolicity?


Example <ref> shows that there are shift operators which are not linearly conjugate to a finite product of particular types of weighted shifts. Yet, the following remains open.

Are there shift operators, determined by sequences of matrices on finite dimensional Hilbert spaces, such that σ_S is not linearly conjugate to a finite product of shifts?


Weighted backward shifts on ℓ_p() does not distinguish Devaney chaos from frequent hypercyclicity <cit.>. It would be interesting to have a negative solution to the following question.

Do the notions of Devaney chaos and frequent hypercyclicity coincide for shift operators on ℓ_p(X)? The answer is yes when X =.
    




    

 §.§ Acknowledgments
MC and PV were partially supported by CMUP, member of LASI, which is financed by national funds through FCT – Fundação para a Ciência e a Tecnologia, I.P., under the projects with references UIDB/00144/2020 and UIDP/00144/2020. MC and PV also acknowledge financial support from the project PTDC/MAT-PUR/4048/2021. PV has benefited from the grant CEECIND/03721/2017 of the Stimulus of Scientific Employment, Individual Support 2017 Call, awarded by FCT. UD would like to thank warm hospitality and partial support of CMUP, at the University of Porto, and Jagiellonian University where he enjoyed support as Jagiellonian Scholar.





siam





Maria Carvalho
 
CMUP & Departamento de Matemática,
 
Faculdade de Ciências da Universidade do Porto, 

Rua do Campo Alegre s/n, 4169-007 Porto, Portugal.

E-mail: mpcarval@fc.up.pt


Udayan B. Darji
 
Department of Mathematics,
 
University of Louisville,

Louisville KY 40292, USA.

E-mail: ubdarj01@louisville.edu


Paulo Varandas
 CMUP & Faculdade de Ciências da Universidade do Porto, 

Rua do Campo Alegre s/n, 4169-007 Porto, Portugal.
 
Departamento de Matemática e Estatística,

Universidade Federal da Bahia,

Av. Ademar de Barros s/n, 40170-110 Salvador, Brazil.

E-mail: paulo.varandas@ufba.br




 Paper Title: Observability of complex systems via conserved quantities.



Multipartite Entanglement for Multi-node Quantum Networks
    G. Araneda
    August 5, 2024
=========================================================




     ^aDepartment of Mathematics,
      Florida State University,
  Tallahassee, FL 32306-4510, USA



     ^bInstitute of Molecular Biophysics,
      Florida State University,
  Tallahassee, FL 32306-4510, USA



     ^cProgram in Neuroscience,
      Florida State University,
  Tallahassee, FL 32306-4510, USA



     ^dDepartment of Mathematics,
      University of Kentucky,
  Lexington, KY 40506-0027, USA


§ ABSTRACT Many systems in biology, physics, and engineering are modeled by nonlinear dynamical systems
where the states are usually unknown and only a subset of the state variables can be physically measured. Can we understand the full system from what we measure? In the mathematics literature, this question is framed as the observability problem. It has to do with recovering information about the state variables from the observed states (the measurements). In this paper, we relate the observability problem to another structural feature of many models relevant in the physical and biological sciences: the conserved quantity. For models based on systems of differential equations,
conserved quantities offer desirable properties such as dimension reduction which simplifies model analysis.
Here, we use differential embeddings to show that conserved quantities involving a set of special variables provide more flexibility in what can be measured to address the observability problem for systems of interest in biology.
Specifically, we provide conditions under which a collection of conserved quantities make the system observable.
We apply our methods to provide alternate measurable variables in models where conserved quantities have been used for model analysis historically in biological contexts.








§ INTRODUCTION

A fundamental question in nonlinear dynamics is whether the entire state of a system can be inferred from measurements of a subset of outputs of the states that comprise the system. In the mathematics literature, this is referred to as the observability problem <cit.>. Briefly, a dynamical system is called observable if one can obtain complete information about the internal state of the dynamical system from measurements of a subset of the outputs.

This question is of central importance in physical and biological applications. Nonlinear dynamical systems have been used to model chemical reaction networks <cit.>, combustion reaction networks <cit.>, power grids <cit.>, biophysical networks <cit.>, epidemics <cit.>, and cancer <cit.>.  Observability of such dynamical systems is vital to constructively inform experimentalists and engineers what should be measured to optimize inference of the progress of their work. Most often, all variables involved are unable to be measured. Determining which outputs of a system should be measured to understand the full system is thus useful and essential for scientific and technological progress across disciplines.

For example, in determining the kinetic properties of an enzymatic reaction, one of the biochemical species must be measured to understand reaction rate. It is a challenging endeavor to simultaneously measure all constituents of the enzymatic reaction. Observable dynamical systems can inform biochemists of which species to track to understand the full system. Similarly, in an infection outbreak, observable dynamical systems can inform epidemiologists of what populations to track to optimally understand the dynamics of an epidemic. It is thus vital to develop mathematical theory and methods to ascertain whether a dynamical system is observable, and, if so, to determine which observables render the dynamical system observable.

Several methods exist to determine whether a system is observable.  A longstanding method is to look at the Lie derivatives of the observables with respect to the governing nonlinear vector field and construct the Jacobian matrix of the Lie derivatives <cit.>. Parameter regimes where the Jacobian has full rank are those where the chosen observable renders the full system observable. Another related approach transforms the phase space of the nonlinear system via a differential embedding and considers the rank of the Jacobian matrix as an indicator of observability <cit.>.  Still another very popular method is to construct the associated graph of the nonlinear system and study the strongly connected component decomposition <cit.>. A central result of this graphical approach is that observing dynamics of the source nodes is necessary and sufficient for observability of the full system.  Finally there are methods based on a strongly positive definite condition and sensor selection based on optimization to ascertain observability <cit.>.

In this paper, we expand on the aforementioned results by considering the effect of another property of dynamical systems that manifests in several biological and physical applications:  the conserved quantity. A conserved quantity of a dynamical system is a function of the state variables that remains invariant in time. Typical uses of a conserved quantity is dimension reduction of the system under scrutiny. Because it defines a dependency between the state variables of system, the dynamics of the full system are contracted to a submanifold of the phase space, thereby potentially simplifying analysis.

We show that conserved quantities in combination with differential embeddings provides a means to identify alternative observables in a system that render a system observable. We emphasize that existing methods for determining observability do not consider conserved quantities or how they impact the observability of a system. Therefore, the available methods are unable to detect alternative variables that render a system observable. For example, we show in this paper with an example that the graphical approach can miss alternate observables imputed by conserved quantities.

Our approach is of interest to experimentalists and engineers because it provides a means to identify system outputs to measure that could reveal the internal state of the process being studied. Current methods for identifying observables may lead to concluding that the only observable is an output that cannot be measured. Our method provides flexibility in such scenarios.

Mathematically, our contribution is to append to the rich literature on observable and controllable systems.  We claim that if system dynamics can be contracted to a submanifold that is inherent to the system, there will be more observables than what previous methods predict. Furthermore, our main result describes conditions for which submanifold to contract dynamics if more than one are inherent to the system.









§ BACKGROUND


We consider dynamical systems of the form

    dx(t)/dt = f(x(t))

with observable variables y = g(x̂), where 
f:ℝ^n→ℝ^n and g:ℝ^m→ℝ^m are differentiable functions. We assume that we can only measure a subset of the state variables represented by x̂∈ℝ^m and the initial state x_0∈Ω_0⊂ℝ^n is unknown.


The system in Eq. (<ref>) is observable if there is a bijection between the initial states in Ω_0 and the set of trajectories of the observed outputs y(t) for t≥0 <cit.>.


In the following, we describe popular approaches for determining observability of a nonlinear system as in Eq. (<ref>). The approach in <cit.> considers a k-dimensional differential embedding Φ:ℝ^n→ℝ^km given by Φ(x) = (g(x̂),ġ(x̂),⋯,g^(k)(x̂)) (derivatives with respect to time). The map Φ is locally invertible at x_0 if the Jacobian has full rank. That is, the map Φ is locally invertible at x_0 if

    (∂Φ/∂x|_x_0) = n.

The system in Eq. (<ref>) is locally observable if and only if Eq. (<ref>) holds <cit.>.

Another approach is the graphical approach. The graphical approach <cit.> associates a directed graph 𝒢 to the system given by Eq. (<ref>), where the nodes of 𝒢 are x_1, ⋯, x_n and there is an edge from x_i to x_j if x_j appears in the differential equation of x_i. We consider the condensation graph of 𝒢 where we collapse the strongly connected components into a node. The graphical approach in <cit.> states that a necessary and sufficient condition for observability of the system in Eq. (<ref>) is to observe the source nodes in 𝒢 and a variable in each strongly connected component of 𝒢. However, those conditions are neither sufficient nor necessary as we will show in the following example.



    SIR models are popular for describing dynamics of an infectious disease and for unveiling key biophysical parameters that govern the transition of a disease from dissipating in a population to persisting in an endemic state <cit.>. Such models are typically composed of three state variables: S representing the number of susceptible individuals in a population, I representing the number of infected individuals, and R representing the number of recovered or removed individuals.  They have been shown to apply to more general settings as well by incorporating spatial and stochastic dynamics in their structure <cit.>. Furthermore, they have been used to study dissemination of information through a social network in a number of studies <cit.>. Hence, SIR models form a crux of much of mathematical epidemiology literature.



One of the simplest SIR models describes the dynamics of an epidemic on a short timescale.  In such instances, the impact infection imparts on population dynamics vastly outweighs birth and death events, so birth and death terms do not manifest in the SIR dynamics. Because of this, the total number of individuals is invariant in time. Such a model is applicable, for example, in describing the dynamics and spread of the flu virus through a population <cit.>.

Here we investigate such a model.  Consider the following SIR model:

    dS/dt   = -β SI 
    dI/dt   = β SI - λ I 
    dR/dt   = λ I ,

where S represents the susceptible population, I represents the infected population, and R represents the recovered population. The parameter β quantifies the infectivity of the infectious disease under consideration; thus, the β SI term captures the rate at which susceptible individuals become infected through contact with infected individuals. The parameter λ quantifies the rate of recovery of an infected individual.  This system contains a conserved quantity, namely the total population. That is, ∀ t, S + I + R = N for a prescribed N ∈ℝ. Because of this conserved quantity, Eq. (<ref>) can be reduced to a two-dimensional system that has the same equilibria and stability as the full system. Such a reduction greatly facilitates analysis.

We depict the associated graph of this model in Fig. <ref>A. According to the graphical approach in <cit.>, it is necessary to measure R to make the system observable and that just measuring I would not make the system observable. However, that conlusion would be wrong as we can get the information for S and R by measuring I and using the conserved quantity N. Indeed, S =  1β( İI + λ) and R=N-I-S = N - I -  1β( İI + λ) expresses both R and S as functions of I, İ = dI/dt, and the conserved quantity N.

    

One could apply the graphical method to

    dS/dt   = -β SI 
    dI/dt   = β SI - λ I 
    dN/dt   = 0

to obtain that observing either I and N (or S and N) suffices to recover all variables, including R=N-I-S, but the measurement of the unchanging quantity N is practically quite different from measuring the varying quantity I.





For m≤ n, a scalar-valued function H:ℝ^m→ℝ is a conserved quantity of Eq. (<ref>) if, for all time and initial conditions,

    dH/dt = 0.


Note that if m=n, then the condition in Eq. <ref> can also be stated as

    ∇ H·dx(t)/dt = ∇ H· f(x(t)) = 0,

where ∇ = (∂/∂ x_1,⋯,∂/∂ x_n).


We can represent ℓ conserved quantities H_1,⋯,H_ℓ by using a function G:ℝ^n→ℝ^ℓ where G = (H_1,⋯,H_ℓ). Note that the Jacobian matrix of G is the zero matrix,

    ∂ G/∂x·dx(t)/dt = ∂ G/∂x· f(x) = 0.






§ RESULTS


For m≤ n, a subset of variables s∈ℝ^m are called sufficient whenever observing these variables makes the system observable.
Next, we consider a partition of the variables in Eq. (<ref>), x = (r,s) where r∈ℝ^n-m and s∈ℝ^m is the set of sufficient variables.

Given a collection of conserved quantities G:ℝ^n→ℝ^ℓ, we describe its Jacobian using the partition above as follows:


    ∂ G/∂x(r,s) = [[   ∂ G_1/∂ r_1(𝐫,𝐬)                  ⋯ ∂ G_1/∂ r_n-m(𝐫,𝐬)   ∂ G_1/∂ s_1(𝐫,𝐬)                  ⋯   ∂ G_1/∂ s_m(𝐫,𝐬);                  ⋮                  ⋱                  ⋮                  ⋮                  ⋱                  ⋮;   ∂ G_ℓ/∂ r_1(𝐫,𝐬)                  ⋯ ∂ G_ℓ/∂ r_n-m(𝐫,𝐬)   ∂ G_ℓ/∂ s_1(𝐫,𝐬)                  ⋯   ∂ G_ℓ/∂ s_m(𝐫,𝐬) ]]
    = [[ ∂ G/∂r(r,s) ∂ G/∂s(r,s) ]]


Now we state our main result.



Let

    {[    ẋ(t)       = f(x(t));       y       =    g(s) ].

be an observable system, where
g:ℝ^m→ℝ^m.
If G:ℝ^n→ℝ^ℓ
is a collection of conserved quantities involving sufficient nodes s and
other variables r where
∂ G/∂s(r,s)
is invertible and ∂ G/∂r(r,s) full rank,
then ∃ĝ:ℝ^n-m→ℝ^m such that the system

    {[    ẋ(t)       = f(x(t));       y       =    ĝ(r) ].

is observable.


Since G consists of conserved quantities,
G=constant.
Then, by the implicit function theorem, there is a function ψ:ℝ^n-m→ℝ^m such that s = ψ(r). Let ĝ = g∘ψ 
Since the system in Eq. (<ref>) is observable, the embedding Φ(x) = (g(s),g'(s),⋯,g^(k)(s)) is injective.
Let Φ̂(z) = (ĝ(r),ĝ'(r),⋯,ĝ^(k)(r)) as illustrated in the following diagram, 



ℝ^n [r, "Φ" ] [rdd, dotted]     ℝ^km [dd, "Φ̂"] 

      

     ℝ^km




Then,

    ∂Φ̂(z)/∂z = ∂Φ(x)/∂x∂Ψ(r)/∂r,
       but   ∂Ψ(r)/∂r = -[∂ G/∂s(r,s)]^-1∂ G/∂r(r,s).

Then,


    ∂ (Φ̂∘Φ)(x)/∂x = ∂Φ̂(x)/∂z ∂Φ(x)/∂x = 
    -[∂ G/∂s(r,s)]^-1∂ G/∂r(r,s)
     ∂Φ(x)/∂x.


Thus, Φ̂∘Φ is one-to-one which makes the system in Eq. (<ref>) is observable.







§ APPLICATIONS

Here we demonstrate that relatively simple systems of interest in biology containing conserved quantities are observable through the lense of Theorem <ref>.



 §.§ Constant Population SIR Model

In the following we first ascertain that Eq. (<ref>) is observable provided the observed state variable is R(t). Then we construct the differential embedding map for the system and show that implementing the conserved quantity allows observing other state variables to render the full system observable.


The system given in Eq. (<ref>) is observable. The observed variable will be R(t). To determine whether or not Eq. (<ref>) is observable with R(t) as the scalar observable, we must look at the Jacobian matrix associated with the Lie derivatives of this system <cit.>.  Writing Eq. (<ref>) compactly as

    d𝐗/dt = f(𝐗)

with 𝐗≡ (S, I, R)^T and f(𝐗) = (-β SI, β SI - λ I, λ I)^T, the Lie derivative of a scalar observable y(t) is given by

    ℒ(y) = ∂ y/∂ t + ∑_i = 1^3 f_i ∂ y/∂𝐗_i

In accordance with the usual computations necessary for ascertaining observability, we compute

    ℒ^0 (R) = R     ℒ^1 (R) = 2λ I     ℒ^2(R) = 4λ (β S - λ)I

and construct the associated Jacobian matrix given by

    𝒥 = ([ ∇ℒ^0(R); ∇ℒ^1(R); ∇ℒ^2(R) ]).

That is, each row of the Jacobian matrix consists of a gradient vector of the Lie derivatives with respect to the state variables of the system. When R(t) is observed, the Jacobian matrix is

    𝒥 = ( [           0           0           1;           0          2λ           0;       4λβ I 4λ(β S - λ)           0 ]),

which has full rank provided λ≠ 0, β≠ 0, and I ≠ 0. Having full rank implies the system is observable.


The corresponding differential embedding is bijective. Consider the embedding Φ(S,I,R) = (R, Ṙ, R̈)^T = (R, λ I, λ(β SI - λ I))^T. This is bijective.


To prove injectivity, let Φ(S_1, I_1, R_1) = Φ(S_2, I_2, R_2). Then

    ([                   R_1;                 λ I_1; λ (β S_1 I_1 - λ I_1) ]) = ([                   R_2;                 λ I_2; λ (β S_2 I_2 - λ I_2) ])

This clearly implies R_1 = R_2 and I_1 = I_2. Finally, we have λ(β S_1I_1 - λ I_1) = λ(β S_2I_2 - λ I_2).  Since I_1 = I_2, this implies S_1 = S_2 and injectivity is proved. For surjectivity, take Φ(S,I,R) = (a,b,c)^T for some (a,b,c)^T in the codomain of Φ. Then clearly we can take R = a, I = b/λ and S = c/λβ b + λ/β as a preimage and surjectivity is proved.

One subtle point is that we must constrain the codomain of Φ to be ℝ^3∖{(a,0,c):a,c ∈ℝ} for it to be surjective. This is completely consistent with the Jacobian in Eq. (<ref>), which says that I ≠ 0 is necessary for observability.  This is also consistent physically, since a situation where I = 0 is not particularly interesting when studying the spread of disease.

With the bijectivity of the differential embedding established, it is sufficient to consider the Jacobian of various embeddings to determine whether or not the observed variable renders the full system observable. From this perspective, we next show that observing I in the absence of the conserved quantity does not render the system observable.

Consider now the differential embedding Ψ(S,I,R) = (I, İ, Ï)^T = (I, β SI - λ I, (β S-λ)^2I - β^2 S I^2)^T. Clearly, Ψ is not injective because the image of a point (S, I, R) is agnostic to the value R takes. 


The conserved quantity renders I a sufficient observable. Consider the same differential embedding Ψ = (I, İ, Ï)^T, but now let I = N - S - R, where we solve for I in the conserved population equation S + I + R = N   ∀ t. The corresponding differential equation system becomes

    dS/dt   = -β S (N - S - R)
    dI/dt   = -Ṡ-Ṙ
    dR/dt   = λ (N - S - R)

The corresponding differential embedding is

    Ψ = ( [                                   I;                     (β S -λ)(N-S-R); (β S - λ)^2(N-S-R) - β^2 S(N-S-R)^2 ])

Then, the resulting Jacobian is

    (∂Ψ/∂𝐗) = ([                          0                          1                          0;               β(N- 2S - R)                          0                    λ - β S;                     F(S,R)                          0 -(β S -λ)^2 + 2β^2S(N-S-R) ] )

where F(S,R) = (β S -λ)(2β(N-S-R)-β S + λ) + β^2(N-S-R)(3S+R-N). Again, provided β≠ 0 and λ≠ 0, (∂Ψ/∂𝐗) has full rank and renders the system observable with the observed variable being I(t).



Relating the two embeddings. Since the system in Eq. (<ref>) is observable, the embedding Φ(S,I,R) = (R,Ṙ,R̈)^T is bijective. Let Φ̂(R,Ṙ,R̈) = Ψ = (I,İ,Ï)^T where I = ψ(S,R) = N-R-S.
Then, Φ̂ is a bijection such that the following diagram commutes.




ℝ^3 [r, "Φ" ] [rdd, dotted]     ℝ^3 [dd, "Φ̂"] 

      

     ℝ^3






  §.§.§ Relating to the Graphical Approach

In summary, the preceding discussion says that Eq. (<ref>) is observable if the observed state is R(t). This is consistent with what is obtained in the corresponding directed graph.

In the directed graph of the original SIR system, the only source node is R (see Figure <ref>A). The graphical approach for determining observability states that observing the source nodes of the directed graph of a system is necessary and sufficient to render the system observable. Consistent with the analysis in the previous section, observing R rendered Eq. (<ref>) observable. Furthermore, in the original system, observing I will not render the system observable as I is not a source node. However, it can be made into a source node by invoking the conserved quantity and transforming the system by setting I = N - S - R (see Figure <ref>B). In the transformed system, I is the only source node, thereby making the system observable by observing I.  We note that if we make the transformation S = N - R - I, then S will become the source node and it will be sufficient to observe S to render the system observable. 


A main takeaway is that the existence of the conserved quantity allows for more flexibility in tracking an epidemic from the perspective of the SIR model. Sans the conserved quantity, one can strictly observe only R, the number of recovered individuals, to understand the full system. Simply observing only S or only I will not do the job.  However, the existence of the conserved quantity says that observing any one of the state variables is sufficient to completely understand the system. Thus, trackers of epidemics have flexibility in measuring the epidemic by observing any one of the subpopulations—whichever one is easiest.







 §.§ Michaelis-Menten Kinetics

The simplest enzyme kinetics are Michaelis-Menten kinetics, applied to enzyme-catalyzed reactions of one substrate and one product <cit.>. An enzyme E binds with its substrate S to form a complex ES which then dissociates into E and P, the product of the enzymatic reaction. The reaction network is as follows:

    E + S <=>[k_1][k_-1] ES ->[k_2] E + P

where k_1, k_-1, k_2 are rate constants quantitating the corresponding reactions. Using the law of mass action, we can derive a model characterizing reaction (<ref>).  Let e ≡ [E], s ≡ [S], c ≡ [ES],  and  p ≡ [P]. Then we have <cit.>

    de/dt   = (k_-1 + k_2)c - k_1es 
    ds/dt   = k_-1c - k_1es 
    dc/dt   = k_1es - (k_-1 + k_2)c 
    dp/dt   = k_2c

There are two conserved quantities in this system:

    e + c = E_0
       s + c + p = S_0

where E_0 ∈ℝ represents the initial amount of enzyme in the system and S_0 ∈ℝ is the initial amount of substrate. The two conserved quantities allow for dimensional reduction of system (<ref>) to a planar system

    ds/dt   = k_-1c - k_1(E_0 - c)s 
    dc/dt   = k_1(E_0 - c)s - (k_-1 + k_2)c

By rescaling s, c, and t and assuming that the concentration of substrate vastly outweighs the concentration of enzyme, we can derive the nondimensionalized system

    dσ/dτ   = -σ + (1 - η + σ)ρ
    εdρ/dτ   = σ - (1+ σ)ρ

where σ≡ k_1 s/(k_-1 + k _2), ρ≡ c/E_0, τ≡ k_1 E_0 t. We define the dimensionless parameters ε≡ E_0k_1/(k_-1 +  k_2) and η≡ k_2/(k_-1 + k_2) with 0 < ε≪ 1. We can thereafter invoke the stationary state approximation <cit.> and project onto the slow manifold <cit.> by assuming ρ = σ(1 + σ)^-1. Substituting this expression into the differential equation for p then yields the classical Michaelis-Menten equation:

    dp/dt = V_maxs/K + s

where V_max≡ k_2 E_0 is the fastest rate possible at which product P can be synthesized and K ≡ (k_-1 + k_2)/k_1 is the dissociation constant.

The derivation and generalization of Eq. (<ref>) to more complicated enzyme-substrate mechanisms are a central focus in the theoretical biochemical literature <cit.>. While such derivations are important for the description of biochemical processes, they do not inform experimentalists of the ramifications of the theoretical models to the experiments themselves.

The conserved quantities in the Michaelis-Menten system confine the 4D dynamics to a two-dimensional submanifold, thereby allotting the desirable property of analytic tractabillity in the system. But what does the conserved quantity imply for experimentalists? Broadly, the existence of a conserved quantity consisting of variables that correspond to sources in the directed graph representation [In the enzyme kinetics section of this paper, we will describe observability strictly through the graphical approach.] increases the number of variables that render the full system observable.

The reaction diagram for system (<ref>) is shown in Figure <ref>A. The product P is the only source, implying that to understand the full system (i.e., to render the system observable), one must observe P. In an experimental setting, the kinetics of a given enzyme are measured and calculated from the observed dynamics of P. In a real setting, if P is easily measurable, then the situation at hand is no problem. However, in many situations, the product P is not directly measurable <cit.>. One must find an alternative to derive the kinetics of the corresponding enzymatic reaction. We demonstrate here that the presence of conserved quantities involving source terms allow for more freedom in observing the system.  We now systematically examine how the conserved quantities given in Eqs. (<ref>) alter the reaction diagram.





  §.§.§ Enzyme Conservation

Let us suppose that we only impose enzyme conservation in the system. How does this alter the reaction diagram? In this case, we set e = E_0 - c, and the Michaelis-Menten system becomes

    de/dt   = -dc/dt
    ds/dt   = k_-1c - k_1(E_0 - c)s 
    dc/dt   = k_1(E_0 - c)s - (k_-1 + k_2)c 
    dp/dt   = k_2c.

Following our formalism for obtaining the corresponding reaction diagram, we obtain the diagram shown in Figure <ref>B. It now has two sources: E and P.  This means that to render the system observable, one must observe the dynamics of both E and P. Although the conserved quantity greatly simplifies mathematical analysis, the existence of this conserved quantity thus complicates the experimental setting. The issue arises because the imparted conserved quantity does not consist of the source from the full system, P.

We note that the reaction diagram would have a similar issue even if we took c = E_0 - e in the conserved quantity.



  §.§.§ Substrate Conservation

Now let us examine what happens when we impart substrate conservation. In this case, we set c = S_0 - s - p, rendering system (<ref>) as


    de/dt   = (k_-1 + k_2)(S_0 - s - p) - k_1es 
    ds/dt   = k_-1(S_0 - s - p) - k_1es 
    dc/dt   = -ds/dt-dp/dt
    dp/dt   = k_2(S_0 - s - p)

The corresponding reaction diagram is given in Figure <ref>D. There is again only one source: C. All other nodes have incoming edges including self loops. The implication here is that now we need only observe C to understand the system. Furthermore, if we had set s = S_0 - c - p instead, the only source in the resulting reaction diagram would be S, meaning we need to only observe S to render the system observable. The experimental implication is that one can observe the dynamics of any of S, C, or P to completely understand the system. Hence, if any of S, C, or P are measurable in a laboratory setting, the system can be understood. Thus, the conserved quantity consisting of the source node vastly expanded the number of state variables the we can measure to render the system completely observable.



  §.§.§ Enzyme and Substrate Conservation

What happens if we impose conservation of both enzyme and substrate? Does this simplify the system further? In this case, we set c = S_0 - s - p and e = E_0 -c = E_0 - S_0 + s + p. The system becomes

    de/dt   = ds/dt + dp/dt
    ds/dt   = k_-1(S_0 - s - p) - k_1(E_0 - S_0 + s + p)s 
    dc/dt   = -ds/dt - dp/dt
    dp/dt   = k_2(S_0 - s - p)

The corresponding reaction diagram is shown in Figure <ref>C. Again, the diagram depicts two source nodes (E and C), implying one must observe both C and E to understand the system. This is, of course, incorrect.

The above analysis brings to light an important point: one must not conclude that theoretical conserved quantities imply positive experimental ramifications. Indeed, if one only analyzed the model with both substrate and enzyme conservation, they would conclude that one must observe two state variables to understand the enzymatic system. Conserved quantities that do not include source node state variables do not inform the observability of the system. The conserved quantity s + c + p = S_0, on the other hand, yields a correct interpretation of observability. Namely, any one of the terms involved in the conserved quantity can be observed to understand the system.





§ CONCLUSIONS

We summarize the main contributions of this manuscript as follows. Most generally, we have proved a theorem conveying that observable dynamical systems with conserved quantities that involve source nodes in the corresponding directed graph representation of the system can be recast so that many more system outputs than originally thought could be observed to render the system observable. We used differential embeddings to prove this. In effect, we generalized the observability criteria provided by the graphical approach and the rank-based approach of differential embeddings.

Our approach has important implications for physical and biological sciences. Namely, we argue that systems with conserved quantities exhibit more flexibility in what must be observed for the full system to be understood. We demonstrate this with two concrete biological examples with conserved quantities:  the constant population SIR model and the classical Michaelis-Menten system for enzymatic reactions. For the former model, the original system necessitates observation of R(t) to render the system observable.  However, the conserved quantity allows any one of S, I, or R to be observed for the system to be observable.  Similarly, the classical Michalis-Menten system requires observation of the product, P(t), to render the system observable. The appropriate conserved quantity allows for product, substrate, or enzyme-substrate complex to be observed for the full system to be understood. Such flexibility can be the difference between success and failure in experimental settings.

For dynamical systems exhibiting multiple conserved quantities, our method identifies the `correct' submanifold of phase space to which dynamics should be contracted to obtain alternative observables that render the full system observable. Only conserved quantities that incorporate source nodes of the associated directed graph of the dynamical system can yield other outputs of the system that render the dynamical system observable.

Mathematically, we contribute to the rich mosaic of literature available on controllable and observable systems. Our method will be of interest because it expands upon and improves the popular methods given by the graphical approach and the rank-based differential embeddings approach.






 
ieeetr



























































































































 Paper Title: On the Eigenvalues of Graphs with Mixed Algebraic Structure.


[†]University of Groningen — Bernoulli Institute for Mathematics, Computer Science and Artificial Intelligence; Groningen, The Netherlands
GalleryGPT: Analyzing Paintings with Large Multimodal Models
    Heng Tao Shen
    Received 9th July 2024; accepted 13th July 2024
============================================================



We study some spectral properties of a matrix that is constructed as a combination of a Laplacian and an adjacency matrix of simple graphs. The matrix considered depends on a positive parameter, as such we consider the implications in different regimes of such a parameter, perturbative and beyond. Our main goal is to relate spectral properties to the graph's configuration, or to basic properties of the Laplacian and adjacency matrices. We explain the connections with dynamic networks and their stability properties, which lead us to state a conjecture for the signature.   









15A18, 15A15, 05C50, 05C70, 05C90, 47A55, 34D20




§ INTRODUCTION


The study of algebraic structures associated with graphs is remarkably relevant for pure and applied mathematics. In the first place, the study of matrices associated with networks can provide information on some features of the associated graph, and vice-versa <cit.>. In addition, in the realm of complex dynamical systems, the linear part of a vector field can be recognised as a matrix related to a network <cit.>. In particular, when studying the stability of the critical points of a vector field, one is especially interested in the eigenvalues of the linearisation at such points. In turn, for matrices associated with graphs, it is often possible, or desirable, to have some connection between the eigenvalues and some graph properties <cit.>. Consequently, we can relate the stability properties of complex dynamical systems to the properties of a graph.

A lot of work has been done to extend the possible algebraic structures associated with a graph, for example, accounting for directed edges or self-loops <cit.>, positive and negative weights <cit.>, hypergraphs <cit.>, etc. In this manuscript, we study graphs whose set of edges is split into two classes. We assign to one class a Laplacian structure and to the other class an adjacency structure. We propose a matrix, M(ϵ), that is a combination of such Laplacian and adjacency matrices, where the parameter ϵ acts as a global weight for the adjacency matrix. Precise definitions will follow in the text. Having in mind the potential application to gradient systems, or Hamiltonian, the Laplacian matrix represents stable interactions, while the adjacency matrix represents unstable saddle-like interactions, see Section <ref>. We exploit some relations between the graphs' properties and their spectrum. In particular, we characterise the perturbative regime to provide conditions for positive definiteness, which in turn can be related to the stability of the aforementioned dynamical systems. Additionally, we provide an upper bound for the number of transitions of the signature of M(ϵ) as the parameter ϵ varies. These transitions are important, among other reasons, due to their relationship to bifurcations in dynamical systems. Motivated by the intuition inferred from the dynamics, we state a conjecture for the monotonic change of signature, which we corroborate with a numerical investigation.

The paper is arranged as follows. First, in Section <ref> we introduce the main object of study together with some fundamental properties relevant for the treatment. Section <ref> is dedicated to perturbative results, while in Section <ref> we study the signature's transitions. Later, in Section <ref> we exploit the connection with networked dynamical systems. In Appendix <ref> we prove a result necessary for a proof (Proposition <ref>) contained in the main text.



§ PRELIMINARIES


Let ℱ={𝒱, ℰ} be a simple graph <cit.>, where 𝒱={ 1, …, n } is the set of vertices, and ℰ is the set of (undirected) edges {i,j}∈ℰ.


    The adjacency matrix, A_ℱ, of a graph  ℱ is the n× n matrix with components
    
    (A_ℱ)_ij :=
            
                1     if {i,j}∈ℰ , 
    
                0     otherwise .





    The Laplacian matrix, L_ℱ, of a graph  ℱ is the n× n matrix with components
    
    (L_ℱ)_ij :=
            (i)     if i=j , 
    
                -1      if {i,j}∈ℰ , 
    
                0     otherwise ,

    where (i) is the degree of the node i, i.e., the number of edges incident to the node i.




    Let us consider the subgraphs 𝒢={𝒱, ℰ_𝒢} and ℋ={𝒱, ℰ_ℋ} of ℱ, such that ℰ_𝒢∩ℰ_ℋ =∅ and ℰ_𝒢∪ℰ_ℋ = ℰ. We define the linear operator

    M(ϵ):= L_𝒢 + ϵ A_ℋ ,

where L_𝒢 is the Laplacian matrix of 𝒢, A_ℋ is the adjacency matrix of ℋ, and ϵ>0 is a positive real parameter.


Notice that ϵ acts as global weight on the adjacency matrix. In particular, in the limit ϵ=0 the matrix M(0) becomes the Laplacian matrix L_𝒢, while for ϵ≫ 1 the adjacency matrix A_ℋ will take the leading role. Such a structure can appear when modelling networked systems with different interactions, in that case the parameter ϵ can be seen as modulating the strength between the two coupling classes. For the entirety of the paper, otherwise explicitly mentioned, the notation relative to the graphs ℱ, 𝒢, ℋ,  and their definitions is given by Definition <ref>.











    Since L_𝒢 and A_ℋ are symmetric n × n matrices, then M(ϵ) is an n × n symmetric matrix. Therefore the eigenvalues μ_1(ϵ), …, μ_n(ϵ) of M(ϵ) are real, and there exists a complete basis of orthogonal eigenvectors.


We can immediately identify a couple of relevant properties of the matrix (<ref>). The first, is that the trace does not depend on the value of the parameter ϵ, but only on the number of edges of 𝒢.

    The trace of M(ϵ) is given by
    
    tr(M(ϵ))= 2 #ℰ_𝒢,

    where #ℰ_𝒢 denotes the number of edges of 𝒢.


    The statement follows straightforwardly from the fact that the adjacency matrix is traceless and that the trace of the Laplacian matrix is twice the number of edges of the associated graph. 


The other property we are going to state is related to the symmetries. We notice that the symmetry properties of the matrix (<ref>) are in correspondence with the symmetries of the graph ℱ, where the edges belong now to two classes associated respectively to 𝒢 and ℋ.


    A nonsingular matrix S is called a symmetry of a matrix M if S^-1MS=M, or equivalently [M,S]=0, where [·,·] denotes the commutator.

The set of matrices S that commute with the matrix M, together with the matrix product, form the symmetry group of M.


    Let ℱ be a simple graph. The automorphism group (ℱ) is the set of permutations leaving the adjacency structure of the graph ℱ invariant.


Let us consider the standard permutation-matrix representation of (ℱ) in ℝ^n, where n is the number of vertices of ℱ. Since this is the only representation we consider in this paper, we do not explicitly distinguish the group and its representation, so we write S ∈(ℱ). As a consequence of Definition <ref> we have that [S, A_ℱ]=0, ∀ S ∈(ℱ). Similarly, the commutation relations hold also for the Laplacian matrix.



    The group (𝒢) ∩(ℋ) is the symmetry group of the matrix M(ϵ).






 Let S ∈(𝒢) ∩(ℋ) then, by definition, we have [S, L_𝒢]=0, [S, A_ℋ]=0. Therefore [S,  L_𝒢 + ϵ A_ℋ] = [S,  L_𝒢 ]+ ϵ [S, A_ℋ]  =0, ∀ S ∈(𝒢) ∩(ℋ), and ∀ϵ.




    The matrix L_𝒢 is semi-positive definite, with ((L_𝒢)) equal to the number of connected components of 𝒢. Let 𝒢_1, …, 𝒢_r be the connected components of 𝒢, then the eigenvectors of the zero eigenvalues are 
    
    1_𝒢_1:=(1,⋯,1_#𝒱_𝒢_1, 0, ⋯, 0)^⊺, 1_𝒢_2:=(0, ⋯,0,1,⋯,1_#𝒱_𝒢_2, 0, ⋯, 0)^⊺, …, 1_𝒢_r:=(0, ⋯,0,1,⋯,1_#𝒱_𝒢_r)^⊺ ,

    where #𝒱_𝒢_i is the number of nodes in the connected component 𝒢_i, i=1, …, r.



    Let 𝒢 be a graph with r connected components. The normalised eigenvectors associated with the zero eigenvalues of L_𝒢 are given by
    
    1̂_𝒢_i:= 1_𝒢_i/√(#𝒱_𝒢_i) ,

    where i =1, …, r.



    The inertia of a matrix Q is the ordered triplet { N_-(Q) , N_0(Q) , N_+(Q) }, where N_-(Q) , N_0(Q) , N_+(Q) are respectively the number of negative, zero and positive eigenvalues of Q counted with multiplicity.



    The signature of a matrix Q is the number s(Q)=N_+(Q) - N_-(Q).



    For nonsingular symmetric matrices, the signature uniquely identifies the inertia and vice-versa. Moreover, thanks to Sylvester's law of inertia <cit.>, the signature and the inertia of a matrix are basis independent.











 §.§ Perturbation theory


In this section, we study the spectral properties of M(ϵ) in the regime of small perturbations, 0<ϵ≪1. Let us expand the eigenvalues of M(ϵ) in asymptotic series, 

    μ_i(ϵ) =  μ_i^(0) + ϵμ_i^(1) + ϵ^2  μ_i^(2) +   ,

where i =1, …, n. Notice that the zeroth-order terms are the eigenvalues of the Laplacian matrix L_𝒢, i.e., μ_i^(0)=λ_i, where λ_i ∈spec(L_𝒢). 


    Let 𝒢 be a graph with n vertices and r connected components. Then there are at least n-r eigenvalues of M(ϵ) that, for ϵ sufficiently small, are positive reals. 



    Since 𝒢 has r connected components, then L_𝒢 has r repeated zero eigenvalues and n-r positive eigenvalues. Thanks to Gershgorin Theorem <cit.>, for ϵ small enough the positive eigenvalues will remain positive.


Let us notice that Proposition <ref> holds for any small symmetric perturbation of L_𝒢. The situation is more delicate when we study the first-order perturbation of the zero eigenvalues of L_𝒢. In such case, it is relevant to consider the actual structure of the perturbation ϵ A_ℋ.



    Let 𝒢 be a graph with r connected components, 𝒢_1, …, 𝒢_r. Let μ_1(ϵ), …, μ_r(ϵ) be the eigenvalues of M(ϵ) with zeroth-order term equal to zero, i.e., μ_1^(0)= … =μ_r^(0)=0. The first-order corrections are μ_1^(1)= ϵθ_1, …, μ_r^(1)=ϵθ_r, where θ_1, …, θ_r are the eigenvalues of the matrix
    
    [             2#ℰ_ℋ^(𝒢_1)#𝒱_𝒢_1 #ℰ_ℋ^(𝒢_1,𝒢_2)√(#𝒱_𝒢_1#𝒱_𝒢_2)                             … #ℰ_ℋ^(𝒢_1,𝒢_r)√(#𝒱_𝒢_1#𝒱_𝒢_r); #ℰ_ℋ^(𝒢_2,𝒢_1)√(#𝒱_𝒢_2#𝒱_𝒢_1)             2#ℰ_ℋ^(𝒢_2)#𝒱_𝒢_2                             … #ℰ_ℋ^(𝒢_2,𝒢_r)√(#𝒱_𝒢_2#𝒱_𝒢_r);                             ⋮                             ⋮                             ⋱                             ⋮; #ℰ_ℋ^(𝒢_r,𝒢_1)√(#𝒱_𝒢_r#𝒱_𝒢_1) #ℰ_ℋ^(𝒢_r,𝒢_2)√(#𝒱_𝒢_r#𝒱_𝒢_2)                             …             2#ℰ_ℋ^(𝒢_r)#𝒱_𝒢_r ] ,

    where #ℰ_ℋ^(𝒢_i) is the number of edges of ℋ connecting nodes of 𝒢_i, and #ℰ_ℋ^(𝒢_i,𝒢_j) is the number of edges of ℋ connecting a node of 𝒢_i to a node of 𝒢_j.


    We need to find the first-order correction for the repeated zero eigenvalues of L_𝒢. Since the matrix L_𝒢 is symmetric we have r distinct orthogonal eigenvectors associated with the zero eigenvalues (<ref>). So, by applying the results of <cit.> for weak interactions, we find that the equation to be solved is 
    
    ( ⟨ A_ℋ1̂_𝒢_i, 1̂_𝒢_j⟩ - θδ_ij) ,

    where i,j = 1, …, r, δ_ij is the Kronecker delta, and ⟨· , ·⟩ is the standard scalar product in ℝ^n. By computing explicitly the matrix elements ⟨ A_ℋ1̂_𝒢_i, 1̂_𝒢_j⟩ we retrieve the matrix (<ref>).




    Let 𝒢 be a graph with two connected components, 𝒢_1 and 𝒢_2. For ϵ sufficently small, the matrix M(ϵ) is positive definite iff
    
    (#ℰ_ℋ^(𝒢_1,𝒢_2))^2< 4 #ℰ_ℋ^(𝒢_1)#ℰ_ℋ^(𝒢_2).

    



    For a graph 𝒢 with two connected components the matrix (<ref>) reduces to
    
    [            2#ℰ_ℋ^(𝒢_1)/#𝒱_𝒢_1 ℰ_ℋ^(𝒢_1,𝒢_2)/√(#𝒱_𝒢_1#𝒱_𝒢_2); ℰ_ℋ^(𝒢_2,𝒢_1)/√(#𝒱_𝒢_2#𝒱_𝒢_1)            2#ℰ_ℋ^(𝒢_2)/#𝒱_𝒢_2 ] =: 
            [ a c; c b ].

    The eigenvalues of (<ref>) are 
    
    θ_1,2 = a+b ±√((a-b)^2 + 4 c^2)/2 .

    Clearly, the first eigenvalue, θ_1= (a+b + √((a-b)^2 + 4 c^2))/2, is always positive as all the terms are positive. The second eigenvalue can be rewritten as θ_2= (a+b - √((a+b)^2 - 4 (ab- c^2)))/2, which is positive only if c^2<ab. The statement follows.
    




    Let 𝒢 be a connected graph and ℰ_ℋ non-empty, then, for ϵ sufficiently small, the matrix M(ϵ) is positive definite. Moreover, the first-order expansion of the zero eigenvalue of L_𝒢 is
    
    μ_0(ϵ) = ϵ2 #ℰ_ℋ/n + O(ϵ^2) .




    The statement follows from the explicit computation of a Rayleigh quotient <cit.>, or simply from the one-dimensional component case of Corollary <ref>.








 §.§ Bifurcation of eigenvalues


When the parameter ϵ is not restricted to small values, the study of the eigenvalues of M(ϵ) becomes more complicated. Indeed, by varying ϵ in the domain of positive reals, one can ask how many times the eigenvalues change sign. Such information is encoded in the inertia and in the signature of the matrix M(ϵ), see Definitions <ref>, <ref>. In the following proposition, we provide an upperbound on the changes of signs for the eigenvalues of M(ϵ).



    Let ϵ>0, and s(M(ϵ)) = N_+(M(ϵ)) - N_-(M(ϵ)) be the signature of the matrix M(ϵ). The maximum number of transitions of signature is
    
    n-((A_ℋ)) .




    The signature can change only if the matrix M(ϵ) is singular for some value of ϵ. In turn, M(ϵ) is singular if (M(ϵ))=0. In order to find an upper bound in the number of transitions of the signature, we need to study the maximal power in ϵ of the polynomial (M(ϵ)). Let M_ij(ϵ) be the components of the matrix M(ϵ), then the determinant can be written as
    
    
    
    
    (M(ϵ)) = ∑_σ∈𝕊_nsgn(σ) ∏_i^n (M(ϵ))_iσ(i) ,

    where 𝕊_n is the symmetric group of permutations over n elements, and sgn(σ) is the sign of the permutation σ. We recall that M(ϵ) is, by definition, M(ϵ)=L_𝒢 + ϵ A_ℋ. So, we have
    
    
    
    
    (M(ϵ)) = ∑_σ∈𝕊_nsgn(σ) ∏_i^n ( (L_𝒢)_iσ(i) + ϵ(A_ℋ)_iσ(i))  .

    We expand the product following the factorisation described in Appendix <ref>, obtaining 
    
    
    
    
    
    
    
    
    
    
    
    
    
    (L_𝒢) +    
        + ∑_σ∈𝕊_nsgn(σ) ( ∑_r=1^n-1ϵ^r C_n,r( (L_𝒢)_1σ(1), …,  (L_𝒢)_nσ(n);  (A_ℋ)_1σ(1), …, (A_ℋ)_nσ(n)) )  + 
       + ϵ^n  (A_ℋ) ,

    where C_n,r is defined in (<ref>), and roughly speaking it is the sum of the non-repeated combination of products where the terms (L_𝒢)_iσ(i) appear n-r times and  (A_ℋ)_jσ(j) appear r times, i≠ j.
    We recall that the Laplacian matrix of a simple graph is singular, so (L_𝒢)=0, and therefore one zero is always at ϵ=0. This fact is easy to check also without the above expansion. In order to have a non-zero term with power r in ϵ it is necessary that at least one term of the form
    
    (A_ℋ)_k_1σ(k_1)⋯(A_ℋ)_k_rσ(k_r) ,

    where k_1, … ,k_r are non-repeated combinations of elements in {1, …, n }, is non-zero. 
    
    
    
    
    For that to be true, the matrix A_ℋ needs to have r linearly independent eigenvectors. So, the maximum power of ϵ is bounded by the number of linearly independent eigenvectors of A_ℋ, which in turn is equal to the number of non-zero eigenvalues of A_ℋ, i.e., n- ((A_ℋ)). 
    



§ QUALITATIVE STUDY OF LINEAR VECTOR FIELDS


The treatment carried out in this paper is also related to the study of networked dynamical systems. Let us assign to each edge of the graph ℱ a potential function depending on the position coordinates associated to the connected vertices, i.e., {i,j}↦ V(q_i, q_j). The potential represents the interaction between the state variables defined on the vertex set. The total potential is given by

    V_ℱ := ∑_{i,j}∈ℰ V(q_i, q_j) .

We consider, for example, gradient vector fields

    q̇ = - ∇_q V_ℱ ,

where ∇_q:=(∂/∂_q_1, …, ∂/∂_q_n)^⊺ is a differential operator. 







The vector field (<ref>) inherits some properties and relations to the graph structure on which it is defined. Following the construction outlined at the beginning of the paper, Section <ref>, we split the interaction into two parts

    V(q_i, q_j) = 
        1/2 (q_i - q_j)^2     if {i,j}∈ℰ_𝒢 , 
    ϵ q_i q_j     if {i,j}∈ℰ_ℋ .


Now, if we explicitly compute equation (<ref>) we get

    q̇   = - M(ϵ) q,

where the matrix M(ϵ) is the one defined in (<ref>), which reads M(ϵ)=L_𝒢 + ϵ A_ℋ. Let us call the interactions of the form (q_i - q_j)^2/2 diffusive, while the interactions of the form q_i q_j we call saddle. Such terminology comes from the dynamical behaviour of the interactions when analysed separately. On the one hand, diffusive interactions give rise to the Laplacian matrix, that represent a linear diffusion process on the graph. For a connected graph, the system subject to diffusion converges to the mean value of the initial conditions. So, the interaction is stable and attracting, with a line of of equilibria representing the thermalisation values. Note that the line of equilibria is given by the eigenspace associated to the zero eigenvalue of the Laplacian. On the other hand, the term saddle is used to classify the behaviour of systems that have both positive and negative eigenvalues. Clearly, the adjacency matrix being traceless has positive and negative eigenvalues. The behaviour close to a saddle is generically unstable, as the only stable lines are given by the eigenspaces associated with the positive eigenvalues of the adjacency matrix.

An interesting and surprising consequence of Corollary <ref>, Lemma <ref>, and Corollary <ref> is that, under some conditions related to the graph structure, the addition of weak, i.e., ϵ≪ 1, saddle interactions can further stabilise the system. As a particular application of Lemma <ref>, we have the following:


 Let ℱ be a simple graph and {𝒢,ℋ} subgraphs of ℱ as defined in Section <ref>. Let 𝒢 have two connected components, namely 𝒢_1={𝒱_𝒢_1,ℰ_𝒢_1} and 𝒢_2={𝒱_𝒢_2,ℰ_𝒢_2} and ℰ_ℋ be non-empty. Then:
 
     
  * If 𝒢_1 or 𝒢_2 is a complete graph, then, for ϵ>0 sufficiently small, the gradient dynamics q̇ = - ∇_q V_ℱ, as defined above, is unstable. 
     
  * Let both 𝒢_i be non-complete graphs. Let s_i be the number of saddle interactions within vertices in 𝒢_i, i=1,2, and s be the number of saddle interactions between vertices in 𝒢_1 and 𝒢_2 (we recall that these saddle interactions are defined through the edge set of ℋ). If s^2<s_1s_2, then, for ϵ>0 sufficiently small, the gradient dynamics q̇ = - ∇_q V_ℱ, as defined above, is stable. 
 





    We emphasise that a similar result as in Proposition <ref> can be stated for more than two connected components, see Corollary <ref>.






Increasing ϵ leads to possible transitions of stability, see Proposition <ref>. Moreover, when ϵ increases the behaviour induced by the adjacency matrix will become more and more dominant. In turn, this would imply an increasing predominance of the unstable interactions associated to the negative eigenvalues of A_ℋ. Essentially, the dimension of the unstable eigenspace grows as ϵ increases. For such reasons we state the following conjecture.


    The signature of the matrix M(ϵ) is a monotonic decreasing function of ϵ.


We corroborate the statement of Conjecture <ref> by a numerical investigation, Figure <ref>. Moreover, we can observe that the conjecture can be extended to any matrix of the form A+ ϵ B, where A is a symmetric real positive semidefinite matrix, and B is a symmetric real matrix, Figure <ref>. Let us notice that if we assume the conjecture to be correct, we can find another bound for the number of transitions of the signature of M(ϵ), complementing the result of Proposition <ref>. In fact, for the limit ϵ→∞ the signature of M(ϵ) becomes the signature of the adjacency matrix A_ℋ, which we denote by s(A_ℋ). In the regime of ϵ≫ 1, we can consider the perturbation problem  A_ℋ + (1/ϵ) L_𝒢. Let us recall that A_ℋ can have some zero eigenvalues that under the perturbation of L_𝒢 will generically perturb to some nonzero real number. Since we look for an upper bound in the number of transitions, we should consider the case where all the zero eigenvalues become negative under perturbation. So, the upper bound is

    n- s(A_ℋ)+((A_ℋ)) .

Then, combining (<ref>) and (<ref>) we have that an upper bound for the number of transitions of signature is given by

    min{ n-((A_ℋ)) ,  n- s(A_ℋ)+((A_ℋ)) } .

The best bound is obtained by examining these two conditions

    if  2 ((A_ℋ)) > s(A_ℋ)   n-((A_ℋ)) ,
       if  2 ((A_ℋ)) < s(A_ℋ)   n- s(A_ℋ)+((A_ℋ)) ,

and, of course, if the equality 2 ((A_ℋ)) = s(A_ℋ) holds then they are both equal. In Figure <ref> we show a comparison between the number of transitions detected and the upper bounds we derived.










§ OUTLOOK


In this paper, we introduced the matrix (<ref>), that we called M(ϵ), representing a mixing between the Laplacian and Adjacency matrix. We studied how some spectral properties are related to the graph's configurations. In turn, we exploited the connection with networks of dynamical systems, in particular with gradient systems. Similarly, one can also derive a connection with Hamiltonian systems, where the stability properties are affected by the symplectic structure. Of course, there are possible generalisations of the definitions we stated. Potential directions include considering other graph structures, e.g., directed graphs, or hypergraphs, but also other combinations of known algebraic structures. It might also be possible to explore in more detail other properties of the matrix M(ϵ), such as its eigenvectors. Moreover, it would be quite significant to prove or further understand Conjecture <ref>.





 












§ FACTORISATION OF THE PRODUCT ∏_I (X_I + Ε Y_I) 


We reserve this appendix for the factorisation in powers of ϵ of the product

    ∏_i=1^n (x_i + ϵ y_i).

The factorisation turns out suitable for the computation of the determinant of M(ϵ), see Section <ref>. In general, the expansion proposed provide the standard polynomial-in-ϵ form for the product (<ref>).


    The product (<ref>) can be rewritten as a polynomial of degree n in ϵ of the form
    
    ∏_i=1^n (x_i + ϵ y_i) = ∑_r=0^n ϵ^r C_n,r(x_1, …, x_n; y_1, …, y_n) ,

    where C_n,r(x_1, …, x_n; y_1, …, y_n) is defined as
    
    C_n,r(x_1, …, x_n; y_1, …, y_n):=  x_1 ⋯ x_n-r y_n-r+1⋯ y_n + … + y_1 ⋯ y_r x_r+1⋯ x_n_nr possible combinations .

    



    We start by expanding (<ref>) as follows
    
    ∏_i=1^n (x_i + ϵ y_i)    = ∑_k_1, …, k_n =0^1 ∏_i=1^n x^k_i (ϵ y)^1-k_i
       = ∑_k_1, …, k_n =0^1 ϵ^n- ∑_i=1^n k_i∏_i=1^n x^k_i y^1-k_i .

    Notice that the exponents k_i and 1-k_i are combinations of 0 and 1. So the sum ∑_k_1, …, k_n =0^1 gives the 2^n combinations of products ∏_i=1^n x^k_i y^1-k_i. We group together the combinations that have the same power of ϵ, i.e., n- ∑_i=1^n k_i =r, r=0,…, n. For a given r we have that there are nr products that are non-repeated combinations of n-r elements of { x_1, …, x_n} and r elements of  { y_1, …, y_n}.















abbrv





§ STATEMENTS AND DECLARATIONS



    
  Funding The authors declare that no funds, grants, or other support were received during the preparation of this manuscript.
    
  Competing Interests The authors have no relevant financial or non-financial interests to disclose.
    
  Data Availability All datasets are available upon request to the corresponding author.





 Paper Title: Asymptotic windings, surface helicity and their applications in plasma physics.









	Wadim Gerner[E-mail address: mailto:wadim.gerner@inria.frwadim.gerner@inria.fr]




Sorbonne Université, Inria, CNRS, Laboratoire Jacques-Louis Lions (LJLL), Paris, France


Abstract: 
In [J. Cantarella, J. Parsley, J. Geom. Phys. 60:1127 (2010)] Cantarella and Parsley introduced the notion of submanifold helicity. In the present paper we investigate properties of surface helicity and in particular answer two open questions posed in the aforementioned work: (i) We give a precise mathematically rigorous physical interpretation of surface helicity in terms of linking of distinct field lines. (ii) We prove that surface helicity is non-trivial if and only if the underlying surface has non-trivial topology (i.e. at least one hole).

We then focus on toroidal surfaces which are of relevance in plasma physics and express surface helicity in terms of average poloidal and toroidal windings of the individual field lines of the underlying vector field which enables us to provide a connection between surface helicity and rotational transform. Further, we show how some of our results may be utilised in the context of coil designs for plasma fusion confinement devices in order to obtain coil configurations of particular "simple" shape.

Lastly, we consider the problem of optimising surface helicity among toroidal surfaces of fixed area and show that toroidal surfaces admitting a symmetry constitute global minimisers.
Keywords: Submanifold helicity, Surface helicity, Biot-Savart operator, Plasma physics, Coil winding surface, Rotational transform
2020 MSC: 14J81, 37E35, 49Q10, 78A30, 78A55


§ INTRODUCTION

	In the realm of 3-dimensional (magneto-)hydrodynamics there is a well-known preserved physical quantity called helicity. To set the stage, we let Ω⊂ℝ^3 be a bounded smooth domain and we let u be a smooth vector field which is divergence-free on Ω and tangent to its boundary. The 3-d Biot-Savart helicity of u may then be defined as follows
	
    ℋ(u):=1/4π∫_Ω×Ωu(x)·(u(y)×x-y/|x-y|^3)d^3yd^3x.

	An alternative way to express helicity consists of defining the Biot-Savart potential of u by 
    BS_Ω(u)(x):=1/4π∫_Ωu(y)×x-y/|x-y|^3d^3y

	so that ℋ(u) is just the L^2-inner product of u and BS_Ω(u), c.f. <cit.>. From a physical perspective helicity may be regarded as a measure of the average linking of distinct field lines of the underlying vector field u, see <cit.>,<cit.>,<cit.>. The relevance of helicity is that it is preserved under the action of volume preserving diffeomorphisms, c.f. <cit.>, <cit.>,<cit.> and in fact is in some sense the only such invariant <cit.>. In the context of the incompressible Euler equations the velocity field v satisfies the equations
	
    ∂_tv+∇_vv=∇ p, div(v)=0 in Ω, v∥∂Ω,

	where p denotes the pressure function. Upon applying the curl operator to the first equation one verifies that the vorticity ω of the fluid, i.e. the curl of the velocity field v, is a solution to
	
    ∂_tω=-[v,ω],
 
	where [·,·] denotes the Lie-bracket of vector fields. This implies that ω(t,x)=((ψ_t)_*ω_0)(x), where ψ_t denotes the flow of v, (ψ_t)_* denotes the pushforward with respect to ψ_t and ω_0(x):=ω(0,x)=curl(v(0,x)) is the initial vorticity. Since v∥∂Ω and div(v)=0 we conclude that ψ_t is well-defined and volume-preserving. We hence conclude from our preceding discussion that the helicity of the vorticity vector field is preserved in time. In the context of ideal magnetohydrodynamics, one can show in a similar spirit that the helicity of the underlying magnetic field is preserved in time. Even more, helicity also provides a lower bound on the L^2-norm of the underlying vector field, which in the context of ideal magnetohydrodynamics corresponds to the magnetic energy of the underlying magnetic field, c.f. <cit.>, <cit.>
	
    |ℋ(u)|≤ c(Ω)∫_Ω|u(x)|^2d^3x

where c(Ω)>0 is a constant independent of u.
It is further interesting to understand which other diffeomorphisms, beyond the volume-preserving diffeomorphisms, preserve helicity. This question has been investigated in <cit.> and it turns out that in general there exist non-volume preserving diffeomorphisms which preserve helicity.

In <cit.> Cantarella and Parsley introduced shortly the notion of submanifold helicity, a special case of which is the surface helicity, which corresponds to the (0,2,3)-helicity in the terminology of <cit.>. As we have seen, the notion of 3-d helicity is an important quantity in the context of fluid dynamics. It is therefore of interest to understand its 2-d analogue since vector fields tangent to surfaces appear naturally in many applications, some of which we will discuss in the next sections.
The main goal of the present manuscript is to illuminate our understanding of surface helicity and establish connections to other known dynamical quantities.
Outline of the paper: In the next section we present the main results which we divided in subsections to increase readability. We first discuss answers to some open questions raised in <cit.> regarding the surface helicity and provide in particular a general interpretation of surface helicity in terms of linkage of distinct field lines of the underlying vector field. In <Ref> we provide an alternative interpretation on toroidal surfaces which connects surface helicity to other important dynamical quantities on the torus. Based on this alternative interpretation we connect in <Ref> the notion of surface helicity to the notion of rotational transform in the context of plasma fusion. We then deal in <Ref> with the question of optimising surface helicity among toroidal surfaces of prescribed area and prove that toroidal surfaces admitting a symmetry are global minimisers while the existence of global maximisers is stated as an open problem. Finally, in <Ref> we discuss how some of the results obtained in the present manuscript can be used to construct "simple" surface currents on a coil winding surface which approximate desired target magnetic fields in a given plasma domain, which is of relevance in stellarator plasma fusion confinement devices. The proofs of the theorems are contained in <Ref>.



§ MAIN RESULTS

Before we state the main results we establish some notation.
Notation: For a given, C^1,1-regular, closed, connected surface Σ⊂ℝ^3 we let L^p(Σ), 1≤ p≤∞, denote the standard space of L^p-integrable functions on Σ and we define the vectorial counterpart by

    L^p𝒱(Σ):={v∈(L^p(Σ))^3| v(x)·𝒩(x)=0 a.e. x∈Σ},

where 𝒩 denotes the outward pointing unit normal on Σ, i.e. the unit normal pointing in the direction of the unbounded component of ℝ^3∖Σ. In addition we let C^0,1𝒱(Σ) denote the Lipschitz-continuous vector fields on Σ which are tangent to Σ at every point. We say that v∈ L^p𝒱(Σ) is divergence-free if ∫_Σ v·∇_Σα dσ=0 for all α∈ C^1(Σ), where ∇_Σ denotes the surface gradient of α which can be computed as ∇_Σα=∇α̃-(𝒩·∇α̃)𝒩, where α̃ is an arbitrary C^1-extension of α to Ω and ∇ denotes the standard Euclidean gradient, and where dσ denotes the standard surface measure. We denote by L^p𝒱_0(Σ) the subspaces of L^p𝒱(Σ) which consist of divergence-free vector fields. Lastly, we say that v∈ L^p𝒱(Σ) is curl-free if ∫_Σv· (∇_Σα×𝒩)dσ=0 for all α∈ C^1(Σ).


 §.§ Properties of Surface helicity

We start by defining the 2-dimensional analogue of the Biot-Savart operator.

	
	Let Σ⊂ℝ^3 be a closed, connected C^1,1-surface. Given 1<p<∞, the following operator is a well-defined, continuous operator called the surface Biot-Savart operator
	
    BS_Σ:L^p𝒱(Σ)→ L^q𝒱(Σ), v↦1/4π(∫_Σv(y)×x-y/|x-y|^3dσ(y))^∥

	where q:=2p/2-p if 1<p<2, where 1≤ q<∞ can be taken arbitrary if p=2 and q=∞ if 2<p≤∞.

In order to define the "tangent part" ·^∥ we cover Σ by any collection of open domains U_k which support local orthonormal frames {E^k_1,E^k_2} of class C^0,1. On each such U_k we define (we drop the superscript k)

    (∫_Σ v(y)×x-y/|x-y|^3dσ(y))^∥:=∑_i=1^2(∫_Σv(y)×x-y/|x-y|^3dσ(y)· E_i(x))E_i(x).

The well-definedness of <Ref> is shown in <Ref>. In particular we show that the defined vector fields agree on the overlaps of the U_k almost surely and that any two covers U_k and Ũ_k induce the same vector field.
We come now to the definition of (cross-)helicity.

	
	Let Σ⊂ℝ^3 be a connected, closed, C^1,1-surface. Then we define the cross-helicity as follows
	
    ℋ_c:L^4/3𝒱(Σ)× L^4/3𝒱(Σ)→ℝ, (v,w)↦∫_Σv(x)·BS_Σ(w)(x)dσ(x)
    		
    
    		=1/4π∫_Σ×Σv(x)·(w(y)×x-y/|x-y|^3)dσ(y)dσ(x).

Further, we define the surface helicity of a vector field by

    ℋ:L^4/3𝒱(Σ)→ℝ, v↦ℋ_c(v,v).



	
	Due to the continuity of BS_Σ:L^4/3𝒱(Σ)→ L^4𝒱(Σ) we see that cross helicity (and therefore helicity) is continuous and that there is some constant c(Σ)>0 such that for all v,w∈ L^4/3𝒱(Σ) we have |ℋ_c(v,w)|≤ c_0v_L^4/3(Σ)w_L^4/3(Σ).

In <cit.> Cantarella and Parsley introduced the notion of helicity of closed forms on (sub-)manifolds of Euclidean spaces. Of particular interest are the (1,3,3)-helicity and the (0,2,3)-helicity which correspond to helicities of 2-forms on domains in ℝ^3 and of 1-forms on embedded surfaces in ℝ^3 respectively. It is shown <cit.> that the (1,3,3)-helicity coincides with the classical helicity of divergence-free fields, tangent to the boundary of a given domain Ω upon an identification of a given closed 2-form α with a vector field v by contraction of the corresponding volume form.

Regarding the (0,2,3)-helicity the authors in <cit.> mention that it should be possible to regard it as a measure of the linkage of distinct field lines of a vector field associated with a given closed 1-form α and they raise the question of whether or not the (0,2,3)-helicity may attain non-trivial values (i.e. whether there exists v∈ C^∞𝒱_0(Σ) with ℋ(v)≠ 0). They conjecture that this should be the case whenever Σ has a non-trivial homology.

To summarise, the following two central problems regarding the (0,2,3) helicity were open <cit.>


	
  * Show that if Σ has non-trivial homology, then the (0,2,3)-helicity is not identically zero.
	
  * Provide a mathematically rigorous physical interpretation of (0,2,3)-helicity.

In the upcoming result we denote by ℋ_(0,2,3) the (0,2,3)-helicity on (closed) 1-forms and by ℋ the (vector field) helicity as introduced in <Ref>. Given some v∈ C^∞𝒱(Σ) we denote by ι_vω_Σ the contraction of the area form ω_Σ via the vector field v. In particular, if v is divergence-free, the associated 1-form is closed. Since the authors in <cit.> deal with smooth domains and surfaces we formulate the following result in the smooth setting.

	
	Let Σ⊂ℝ^3 be a smooth, connected, closed surface and let v∈ C^∞𝒱_0(Σ). Then
	
    ℋ_(0,2,3)(ι_vω_Σ)=ℋ(v).


We exploit this relationship to provide an answer to both of the open problems raised in <cit.> regarding (0,2,3)-helicity.

The following provides an answer to the first open problem regarding surface helicity.

	
	Let Σ⊂ℝ^3 be a closed, connected C^1,1-surface. Then there exists some v∈ L^2𝒱_0(Σ) with ℋ(v)≠ 0 if and only if g(Σ)≥ 1, where g(Σ) denotes the genus of Σ. Further, if v∈ L^2𝒱_0(Σ) is co-exact, i.e. it can be written as v=grad_Σ(f)×𝒩, for some function f∈ H^1(Σ) and where 𝒩 denotes the outward unit normal, then ℋ_c(v,w)=0 for all w∈ L^2𝒱_0(Σ).

Regarding the second question posed above we intend to follow the ideas of <cit.> and <cit.> regarding the physical interpretation of the standard 3-d helicity (which corresponds to (1,3,3)-helicity in the notation of <cit.>).

Note that in general Σ might contain a large "flat region" (a copy of a large 2-d disc) and then if the field lines of a given vector field v∈ C^0,1𝒱_0(Σ) are not necessarily periodic, we might attempt, in the same spirit as in <cit.>, to close the field lines by geodesics. However, at least as long as the endpoints of the considered field lines are contained in the flat region, the geodesics will be lines passing through the corresponding endpoints of the distinct field line segments. But then these artificially added line segments will intersect with "high probability" (since the directions of the lines passing through the pairs of endpoints will be "generically" not parallel). Thus, one should expect that there is a set of positive measure in Σ×Σ for which the "artificially" closed field lines will intersect if we close field line segments by means of geodesics. In addition, the artificially added parts may intersect the original field line segments, see <Ref>.


Therefore it is necessary to find a new way to appropriately close the field lines of the underlying vector field to be able to obtain an interpretation of helicity in terms of average linking numbers.

Given some v∈ C^0,1𝒱_0(Σ) and x∈Σ we let γ_x(t) denote the (unique) field line of v starting at x. Now if we are given some x∈Σ and T>0 it can happen that γ_x(T)=x=γ_x(0) in which case the field line γ_x is closed (possibly a point). In general however we will find γ_x(T)≠ x=γ_x(0) and therefore it is necessary to close the field line in a suitable way to give an interpretation of helicity in terms of linking of suitable closed curves. It is well-known that, if Σ∈ C^2, there exists some τ_0(Σ)>0 such that the map Ψ_τ(x):=x+τ𝒩(x), x∈Σ, is a diffeomorphism onto its image for every |τ|<τ_0 where 𝒩 denotes the outward pointing unit normal, <cit.>. We define Σ_τ:=Ψ_τ(Σ) so that in particular Σ_0=Σ. Now it is the case that for almost every (x,y)∈Σ×Σ the images of the corresponding field lines γ_x(ℝ) and γ_y(ℝ) are disjoint. Given T,S>0 we then consider the curves γ_x:[0,T]→Σ and γ_y:[0,S]→Σ. In order to close them (in case they are not already closed) we proceed as follows: We connect γ_x(T) with Ψ_τ(γ_x(T)) for a suitably chosen τ (which we will be able to choose independent of x,y,T and S) by following the path Ψ_t(γ_x(T)) from t=0 until t=τ. Similarly, we can connect Ψ_τ(x) and x=γ_x(0) by inverting the path Ψ_t(x) from t=0 until t=τ. We note that Ψ_τ(γ_x(T)),Ψ_τ(x)∈Σ_τ and that we are left with connecting Ψ_τ(γ_x(T)) and Ψ_τ(x). We connect these two points by any curve contained in Σ_τ which is of unit speed and whose length can be uniformly bounded independently of x and T. For instance, we can connect these two points by any (length minimising) geodesic on Σ_τ. We close the curve γ_y:[0,S]→Σ in a similar fashion, but in contrast we connect y and γ_y(S) with their corresponding points on Σ_-τ and connect the corresponding two points on Σ_-τ by a geodesic on Σ_-τ. Let us denote these two closed curves obtained in that way by σ^τ_x,T and σ^-τ_y,S, see <Ref> for an illustration. By construction these two closed curves intersect if and only if γ_x([0,T]) and γ_y([0,S]) intersect, which we will show does not happen for almost every (x,y)∈Σ×Σ.
Note also that we explicitly leave the surface Σ to close our field lines.
By convention we set the linking number of a closed curve and any point not lying on this curve to zero.




With these preliminary observations we obtain the following result where we assume the surface Σ to be of class C^2 to avoid technicalities.

	
	Let Σ⊂ℝ^3 be a closed, connected C^2-surface. Then there exists some τ(Σ)>0 such that for every v∈ C^0,1𝒱_0(Σ) and every sequences (T_n)_n,(S_n)_n⊂ (0,∞) which are strictly increasing and diverging to ∞, the functions 
	
    Σ×Σ→ℝ, (x,y)↦lk(σ^τ_x,T_n,σ^-τ_y,S_n)/T_nS_n

	are well-defined elements of L^1(Σ×Σ) for every n and converge to a limit function in L^1(Σ×Σ). Further, we have the identities
	
    ℋ(v)=lim_n→∞∫_Σ×Σlk(σ^τ_x,T_n,σ^-τ_y,S_n)/T_nS_ndσ(x)dσ(y)=∫_Σ×Σlim^L^1_n→∞lk(σ^τ_x,T_n,σ^-τ_y,S_n)/T_nS_ndσ(x)dσ(y).


Here lim^L^1 indicates that we consider the L^1-limit (and not the pointwise limit which exists only upon passing to appropriate subsequences). Therefore surface helicity may be regarded as either the "asymptotic average linking number of distinct field lines of v" or, as is done more commonly in the 3-d case, the "average asymptotic linking number of distinct field lines of v".

	
	If all field lines of v are periodic and there exists some τ>0 such that all field lines of v have a period of at most τ, then we can instead close the field line segments of v by continuing to follow the flow of v until we return to the respective starting points. In this periodic situation no additional technical issues arise by assuming that Σ∈ C^1,1.



 §.§ Surface helicity on toroidal surfaces


We have seen in <Ref> that on an arbitrary (closed) surface Σ, we can interpret helicity as a measure of entanglement of distinct field lines of the underlying vector field. In the 3-dimensional situation entanglement of distinct field lines does not allow to infer information about the dynamical properties of individual field lines. However, in the 2-dimensional situation closed field lines are co-dimension 1 submanifolds and therefore the dynamics of individual field lines may very well influence the ability of the remaining field lines to link. To illustrate this point further, we imagine we are given a toroidal surface Σ≅ T^2 in ℝ^3 and a (regular) vector field v tangent to Σ. We suppose that v has a closed field line γ which wraps once in poloidal direction around Σ. Since distinct field lines do not cross we infer that the remaining field lines of v must be contained within Σ∖γ. But we can continuously "unfold" Σ∖γ and we see that in fact the remaining field lines may be viewed as being contained in a cylinder so that no linking may take place, c.f. <Ref>. Consequently, the helicity of v must be zero by means of the linking interpretation of helicity, <Ref>. We therefore see that the dynamics of individual field lines may have a strong influence on surface helicity and one may thus expect that an alternative interpretation of surface helicity in terms of averages of dynamical properties of individual field lines should be possible.




We focus here on C^1,1-surfaces Σ⊂ℝ^3 which bound solid tori, i.e. Σ=∂Ω for a domain Ω⊂ℝ^3 which is diffeomorphic to the solid torus D^2× S^1, where D^2 is the unit disc in ℝ^2 and S^1 is its boundary. We note that these type of surfaces are the most relevant in plasma physics, as they appear as the boundaries of plasma domains or model the surfaces along which the coils wind which generate the plasma confining magnetic field <cit.>.

Before we state the main result of this subsection let us introduce some notions.

	
	Let T^2≅Σ⊂ℝ^3 bound a C^1,1-solid torus Ω⊂ℝ^3, i.e. Ω≅ D^2× S^1 and ∂Ω=Σ∈ C^1,1. We say that a simple closed C^1-curve σ_p on Σ is purely poloidal if σ_p is contractible when viewed as curve within Ω, but represents a non-trivial element of the first fundamental group of Σ. We call a simple closed C^1-curve σ_t in Σ toroidal if it generates the first fundamental group of Ω once viewed as a curve in Ω.

We note first that, up to orientation and a standard identification, any two purely poloidal curves are path-homotopic within Σ. Similarly, up to orientation, all toroidal curves are homotopic within Ω. However, two toroidal curves may not be (even up to orientation) homotopic within Σ. In addition, any choice of σ_p and σ_t forms a pair of generators of the first fundamental group of Σ.

To motivate the next definition we consider an arbitrary simple closed C^1-curve σ contained in Σ and for a fixed pair σ_p and σ_t of a purely poloidal and a toroidal curve we can express σ=Pσ_p⊕ Qσ_t where ⊕ denotes the concatenation of curves and P,Q∈ℤ denote the amount of times the curves σ_p and σ_t are traversed. We observe first that there is a unique pair γ_p,γ_t∈ℋ(Σ):={v∈ W^1,2𝒱(Σ)|curl_Σ(v)=0=div_Σ(v)} which forms a basis of ℋ(Σ) and satisfies

    ∫_σ_pγ_t=0=∫_σ_tγ_p, ∫_σ_pγ_p=1=∫_σ_tγ_t.

We observe further, which for example follows from an identical reasoning as in <cit.>, that in fact ℋ(Σ)⊂⋂_1<p≤∞W^1,p𝒱(Σ)⊂⋂_0<α<1C^0,α𝒱(Σ) and therefore the line integrals in (<ref>) make sense in a classical sense. Now since σ is homotopic to Pσ_p⊕ Qσ_t, we find

    ∫_σγ_t=P∫_σ_pγ_t+Q∫_σ_tγ_t=Q

and further

    lim_T→∞1/T∫_σ[0,T]γ_t=Q/τ,

where τ>0 is the period of σ and σ[0,T] is the path [0,T]→Σ, t↦σ(t) (which is not simple for T≥τ). Similarly we find P=∫_σγ_p and lim_T→∞1/T∫_σ[0,T]γ_p=P/τ. This motivates the following definition.

	
	Let T^2≅Σ⊂ℝ^3 bound a C^1,1-solid torus Ω⊂ℝ^3. Given a toroidal curve σ_t and a purely poloidal curve σ_p we define the weighted asymptotic toroidal windings resp. weighted asymptotic poloidal windings of a (not necessarily periodic) C^1-curve σ:[0,∞)→Σ by
	
    q̂(σ):=lim_T→∞1/T∫_σ[0,T]γ_t, p̂(σ):=lim_T→∞1/T∫_σ[0,T]γ_p.



	
	We note that γ_t in (<ref>) is independent of the chosen purely poloidal curve σ_p and it only depends on the toroidal curve σ_t via its orientation, i.e. if two toroidal curves are oriented such that they are homotopic within Ω then the corresponding γ_t will be the same and if they are oriented in opposite direction, then the corresponding γ_t will differ by a minus sign. Consequently, q̂(σ) also depends only on the chosen orientation of the toroidal curve. However, γ_p will in general depend on the homotopy type of σ_t when viewed as a curve in Σ so that even if two toroidal curves are homotopic within Ω, the corresponding γ_p may be distinct (but will be the same if the toroidal curves are homotopic within Σ).
		
The following is the natural generalisation to vector fields.

	
	Let T^2≅Σ⊂ℝ^3 bound a C^1,1-solid torus Ω⊂ℝ^3. Given a toroidal curve σ_t, a purely poloidal curve σ_p and a vector field v∈ C^0,1𝒱_0(Σ), we define the weighted asymptotic toroidal windings and the weighted asymptotic poloidal windings of v at a point x∈Σ as follows
	
    q̂(x):=q̂(γ_x), p̂(x):=p̂(γ_x),

	where γ_x denotes the field line of v starting at x. We further define the average weighted asymptotic toroidal windings and the average weighted asymptotic poloidal windings of v by
	
    Q(v):=∫_Σq̂(x)dσ(x)/|Σ|, P(v):=∫_Σp̂(x)dσ(x)/|Σ|,

	where |Σ| denotes the area of Σ.

We note that in order to compute the average one should have q̂,p̂∈ L^1(Σ). This however follows from standard ergodic theoretical arguments. We state it as a separate lemma since it is of importance when establishing a connection between surface helicity and toroidal and poloidal averages.

	
	Let T^2≅Σ⊂ℝ^3 bound a C^1,1-solid torus Ω⊂ℝ^3. Let σ_p and σ_t be a given purely poloidal and a toroidal curve respectively. If v∈ C^0,1𝒱_0(Σ), then q̂,p̂∈ L^1(Σ) and we have
	
    Q(v)=∫_Σv(x)·γ_t(x)dσ(x)/|Σ|, P(v)=∫_Σv(x)·γ_p(x)dσ(x)/|Σ|.



	One natural way to define purely toroidal simple closed curves is to demand that a given toroidal loop is trivial within ℝ^3∖Ω, see also <cit.>. Another useful way to choose a toroidal curve σ_t can consist in making sure that the resulting induced element γ_p is L^2(Σ)-orthogonal to γ_t so that the quantities Q(v) and P(v) essentially correspond to the L^2(Σ)-orthogonal projection of v onto the space ℋ(Σ) in the sense that π(v)=|Σ|(Q(v)γ_t/γ_t^2_L^2(Σ)+P(v)γ_p/γ_p^2_L^2(Σ)) where π denotes the projection onto ℋ(Σ) and v∈ C^0,1𝒱_0(Σ). Both choices can have advantages in distinct situations and therefore we allow general toroidal curves in our considerations.

To establish a connection between the quantities Q and P, which make statements about the average behaviour of individual field lines, and helicity which is a measure of the average linking of distinct field lines, we need to introduce one final notion.
We define the space ℋ_N(Ω) of harmonic Neumann fields on a bounded C^1,1-domain Ω⊂ℝ^3 as follows

    ℋ_N(Ω):={Γ∈ H^1(Ω,ℝ^3)|curl(Γ)=0=div(Γ), Γ∥∂Ω}.

We note first, c.f. <cit.>, that ℋ_N(Ω)⊂⋂_1≤ p<∞W^1,p𝒱(Ω) and thus, due to Sobolev embeddings ℋ_N(Ω)⊂⋂_0<α<1C^0,α𝒱(Ω). Since curl(Γ)=0 it follows that the restriction Γ|_Σ satisfies curl_Σ(Γ|_Σ)=0 in the weak sense and according to the Hodge-decomposition theorem <cit.> we can therefore write

    Γ|_Σ=∇_Σα+γ

for suitable α∈ H^1(Σ) and γ∈ℋ(Σ). Here γ is the L^2-orthogonal projection of Γ|_Σ onto ℋ(Σ). We note that since ℋ_N(Ω) is L^2(Ω)-orthogonal to the gradient fields and because Γ=curl(A) for a suitable vector potential A (for instance A may be taken to be the Biot-Savart potential of Γ) we must have γ≠ 0 whenever Γ≠ 0.

We are now finally ready to give an alternative interpretation of surface helicity in terms of the "dynamics of individual field lines".

	
	Let T^2≅Σ⊂ℝ^3 bound a C^1,1-solid torus Ω⊂ℝ^3. Let σ_p be a purely poloidal curve and σ_t be a toroidal curve. Further, fix any Γ∈ℋ_N(Ω) such that the L^2(Σ)-orthogonal projection γ of Γ|_Σ onto ℋ(Σ) is L^2(Σ)-normalised. In addition let γ̃:=γ×𝒩. Then, given any v∈ C^0,1𝒱_0(Σ), we have
	
    ℋ(v)/|Σ|^2=Q(v)P(v)∫_σ_tγ∫_σ_pγ̃+[ℋ(γ)(∫_σ_tγ)^2+∫_σ_tγ∫_σ_tγ̃]Q^2(v).

	If, moreover, σ_t bounds a bounded C^1,1-surface 𝒜 outside of Ω, i.e. 𝒜⊂ℝ^3∖Ω and ∂𝒜=σ_t, then
	
    ℋ(v)/|Σ|^2=Q(v)P(v)∫_σ_tγ∫_σ_pγ̃.



	
	We observe that (ℋ_N(Ω))=1 whenever Ω is a solid torus, <cit.>. Hence the condition of L^2(Σ)-normalisation of γ determines Γ uniquely up to a minus sign. If a chosen Γ is replaced by -Γ we see that γ as well as γ̃ both are also changed by a minus sign and hence (since ℋ(-γ)=ℋ(γ)) the above formula is independent of that choice, as one would expect. Similarly, if σ_p and σ̃_p are purely poloidal curves on Σ then either they are homotopic and thus the integrals involving σ_p are unchanged or otherwise σ_p and σ̃_p are oppositely oriented, in which case ∫_σ̃_pγ̃=-∫_σ_pγ̃, but P(v) also changes sign (since it is defined via γ_p which in turn is defined via σ_p). In addition, γ_t is independent of the chosen σ_p and thus Q is also independent of that choice. Therefore the right hand side of (<ref>) is independent of the chosen purely poloidal curve. Finally, we observe that Γ|_Σ and γ differ only by a gradient field such that ∫_σ_tγ=∫_σ_tΓ from which it follows that ∫_σ_tγ only depends on the orientation of σ_t (considered as a curve in Ω) and it is clear once again that changing the orientation of σ_t leaves the right hand side of (<ref>) invariant. Since changing orientation of σ_t leaves the right hand side invariant, we are left with understanding how the quantities ∫_σ_tγ̃, P(v) and Q(v) change if we replace σ_t by a toroidal curve σ̃_t which is homotopic to σ_t in Ω. It follows easily from (<ref>) that γ_t and hence Q(v) remains unchanged. On the other hand, since σ_t and σ_p form a basis of the fundamental group of Σ, we may express σ̃_t=σ_t⊕ Pσ_p, where we used that we may assume that σ̃_t is homotopic to σ_t. Then ∫_σ̃_tγ̃=∫_σ_tγ̃+P∫_σ_pγ̃ and one can verify that γ̃_p-γ_p=-Pγ_t so that it follows from <Ref> that P(v)-P(v)=-PQ(v) (where γ̃_p, P(v) are computed with respect to σ_p and σ̃_t). We hence see that the additional terms appearing through the change of P(v) and ∫_σ_tγ̃ when σ_t is replaced by σ̃_t cancel each other. We conclude that the right hand side of (<ref>) is independent of the chosen curves σ_p and σ_t, as it should be because the left hand side of (<ref>) is of course independent of any such choices.



 §.§ Surface helicity and rotational transform


Assume once more that T^2≅Σ⊂ℝ^3 bounds a C^1,1-solid torus Ω⊂ℝ^3 and suppose that we fixed some purely poloidal curve σ_p and a toroidal curve σ_t on Σ. Now, if σ:ℝ→Σ is any other closed C^1-curve, then we can write it as σ=Pσ_p⊕ Qσ_t, where Q may be viewed as the "toroidal" turns of σ and P may be viewed as the "poloidal" turns of σ. We recall that according to (<ref>) we can write

    Q/τ=lim_T→∞1/T∫_σ[0,T]γ_t, P/τ=lim_T→∞∫_σ[0,T]γ_p,

where γ_p,γ_t are determined by the equations in (<ref>). Consequently, if we are interested in the ratio of "poloidal" twists by "toroidal" twists P/Q, we obtain the identity (assuming Q≠ 0)

    ι(σ):=P/Q=lim_T→∞1/T∫_σ[0,T]γ_p/lim_T→∞1/T∫_σ[0,T]γ_t=p̂(σ)/q̂(σ),

where p̂(σ),q̂(σ) denote the "asymptotic windings" of σ. These observations motivate the following definition.

	
	Let T^2≅Σ⊂ℝ^3 bound a C^1,1-solid torus Ω⊂ℝ^3. Let σ_p and σ_t be a given purely poloidal and a toroidal curve respectively and let v∈ C^0,1𝒱(Σ). If q̂(x)≠ 0 for a given x∈Σ, c.f. <Ref>, we define the rotational transform of v at x as
	
    ι_v(x):=p̂(x)/q̂(x),

provided the limits in the definition of p̂(x) and q̂(x) exist.


	
	
		
  * According to <Ref>, if v∈ C^0,1𝒱_0(Σ), then p̂(x) and q̂(x) exist for a.e. x∈Σ w.r.t. the standard surface measure.
		
  * We observe that the definition of the rotational transform depends on the choice of σ_p and σ_t in the following way: We recall first the defining equations (<ref>) of γ_p and γ_t
		
    ∫_σ_pγ_t=0=∫_σ_tγ_p, ∫_σ_pγ_p=1=∫_σ_tγ_t.

		If σ̃_p is any other closed, purely poloidal curve, then either σ̃_p is homotopic to σ_p or otherwise it is homotopic to -σ_p (the curve obtained from σ_p by inverting the orientation). In both cases γ_t remains unchanged, while γ_p changes sign and so q̂ remains unchanged, while p̂ changes sign, i.e. at most the sign of ι is changed, depending whether σ̃_p is oriented in the same way as σ_p or oppositely oriented. On the other hand, if we change σ_t to some σ̃_t, then by definition either σ̃_t will be homotopic to σ_t or -σ_t as a curve in Ω (but not necessarily within Σ). Since σ_p and σ_t generate the first fundamental group of Σ, we can then however express σ̃_t=±σ_t⊕ Pσ_p for a suitable P∈ℤ, where the sign in ± depends on whether σ̃_t is homotopic to σ_t or -σ_t within Ω. Consequently, we see that γ_t at most changes sign, while γ̃_p=γ_p∓ Pγ_t. Thus, q̂ may change its sign, depending whether σ_t and σ̃_t are oriented in the same, or opposite, way within Ω and p̂̃̂(x)=p̂(x)∓ Pq̂(x), where p̂̃̂ indicates that this quantity is computed with respect to the curve σ̃_t. In conclusion ι̃_v(x)=±ι_v(x)∓ P, where once more the tilde indicates that ι_v is computed with respect to the toroidal curve σ̃_t. In particular, if σ_p,σ̃_p are two purely poloidal curves of the same orientation and σ_t,σ̃_t are two poloidal curves of the same orientation (within Ω), then ι̃_v(x)=ι_v(x)+β for some constant β∈ℤ, which is independent of x and the vector field v under consideration. Hence, if we wish to study a problem which aims to maximise ι (possibly under some additional constraints), then the maximisers will be independent of the chosen curves (as long as they are oriented in the same way). Finally, also notice that the property q̂(x)≠ 0 at a given x∈Σ is independent of the chosen curves.
		

We will now first state the main mathematical result of this section, before we shortly discuss the relevance of rotational transform in the context of plasma fusion confinement devices.

In order to do so we state here the (standard) definition of rectifiability of a vector field on a 2-torus T^2=ℝ^2ℤ^2.

	
	Let T^2≅Σ be a C^1,1-surface, which is diffeomorphic to the 2-torus. A vector field v∈ C^0,1𝒱(Σ) is said to be rectifiable if there exists a diffeomorphisms ψ:Σ→ T^2 such that ψ_*v=a∂_1+b∂_2 for suitable constants a,b∈ℝ and the standard coordinate fields ∂_1,∂_2 on T^2, where ψ_*v denotes the corresponding pushforward vector field.

With this we can formulate the main result of this subsection.

	
	Let T^2≅Σ⊂ℝ^3 bound a C^1,1-solid torus Ω⊂ℝ^3. Let σ_p and σ_t be a given purely poloidal and a toroidal curve on Σ respectively. Further, let γ_p,γ_t be a basis of ℋ(Σ) determined by the condition (<ref>). Assume that v∈ C^0,1𝒱(Σ) is rectifiable and that there exists some f∈ C^0,1(Σ,(0,∞)) such that w:=fv∈ C^0,1𝒱_0(Σ), i.e. div_Σ(w)=0. If Q(w)≠ 0, then the rotational transforms of v and w are well-defined at a.e. x∈Σ and there holds
	
    ι_v(x)=ι_w(x)=∫_Σw·γ_pdσ/∫_Σw·γ_t dσ=P(w)/Q(w),

	where the subscript indicates whose rotational transform is considered. In particular, ι_v(x) is independent of x.


	
	In general, for a given vector field v∈ C^0,1𝒱(Σ), even if the rotational transform ι_v(x) is an element of L^1(Σ), it is not true that 1/|Σ|∫_Σι(x)dσ(x)=P(v)/Q(v) because the average of a quotient does not need to coincide with the quotient of the respective averages.

As an immediate consequence of <Ref> and <Ref>, we obtain the following relation between surface helicity and rotational transform.

	
	Let T^2≅Σ⊂ℝ^3 bound a C^1,1-solid torus Ω⊂ℝ^3. Let σ_p be a purely poloidal curve on Σ and let σ_t be a toroidal curve on Σ which bounds a bounded C^1,1-surface 𝒜 outside of Ω. Assume further that v∈ C^0,1𝒱(Σ) is rectifiable and that there exists some f∈ C^0,1(Σ,(0,∞)) such that w:=fv∈ C^0,1𝒱_0(Σ), i.e. div_Σ(w)=0. If Q(w)≠ 0, then
	
    ℋ(w)/|Σ|^2=ι_wQ^2(w)(∫_σ_tγ)(∫_σ_pγ̃),

	where ι_w is the constant rotational transform of w and γ,γ̃ are defined as in <Ref> and in particular are independent of w.


	
	
		
  * We point out shortly that a corresponding formula for the vector field v does not need to hold necessarily, since we require the divergence-free property in order to invoke <Ref>, which is the reason why the conclusion is solely formulated for the vector field w.
		
  * It is also customary to call a vector field v∈ C^0,1𝒱(Σ) semi-rectifiable, provided there exists a (strictly) positive C^0,1-function f such that fv is rectifiable, c.f. <cit.>. Hence <Ref> can be reformulated by assuming that a given w∈ C^0,1𝒱_0(Σ) is semi-rectifiable.
	


	
	In the realm of plasma physics one is interested in steady solutions of the equations of ideal magnetohydrodynamics with a resting plasma <cit.>. Such plasma equilibria are determined by the PDE
	
    B×curl(B)=∇ p and div(B)=0 in 𝒫, B∥∂𝒫

	where 𝒫 denotes the region containing the plasma ("plasma region"), B is the corresponding magnetic field and p is the pressure. According to Arnold's structure theorem, <cit.>, <cit.>, if B is analytic and B and curl(B) are not everywhere collinear, the domain 𝒫 will decompose, after removing a 2-dimensional analytic subset from it, into a finite number of chambers, each of which is invariant under the flow of B and is one of the following two types:
	
		
  * Either the chamber foliates further into toroidal surfaces, each of which is again invariant under the flow of B and such that on each such surface B is rectifiable and hence in particular either all field lines of B are closed (with the same period) or all of the field lines of B are dense within the surface.
		
  * Or the chamber foliates into cylindrical surfaces diffeomorphic to S^1×ℝ, each of which is invariant under the flow of B and such that all field lines of B are closed on any such surface.
	
	Of particular interest are the invariant toroidal surfaces. In particular, in order to counter the plasma particle drift in a fusion plasma it is desirable to have a high rotational transform <cit.>, see also <Ref> for a precise relationship between the different notions of rotational transform. We therefore wish to understand the quantity ι_B on regular toroidal level sets of the pressure p. We note that if Σ is a regular toroidal level set of p, then |∇ p|≠ 0 on all of Σ and in fact, letting f:=1/|∇ p| one can verify that B:=fB is a divergence-free field on Σ (divergence-free with respect to the surface measure on Σ), c.f. <cit.>. It follows therefore from <Ref> that we can compute the rotational transform of B on a given regular toroidal level set of Σ as
	
    ι_B(x)=∫_ΣB·γ_pdσ/∫_ΣB·γ_tdσ

	for a fixed purely poloidal curve σ_p and a toroidal curve σ_t bounding a surface outside of the domain enclosed by the level set and their corresponding induced harmonic fields γ_p,γ_t∈ℋ(Σ). Note once more, that the rotational transform of B (on any fixed regular toroidal surface) is constant.



 §.§ Surface helicity optimisation


In <cit.> Cantarella, DeTurck, Gluck and Teytel considered the following optimisation in 3-dimensional space: First, given a smooth, bounded domain Ω⊂ℝ^3, they show that there exists some smooth vector field v on Ω with div(v)=0, v∥∂Ω such that

    ℋ_3D(v)/v^2_L^2(Ω)=sup_w≠ 0
    div(w)=0
     w∥∂Ωℋ_3D(w)/w^2_L^2(Ω)=:λ(Ω),

where ℋ_3D denotes the standard 3-dimensional Biot-Savart helicity, see <cit.> for a proof of the above fact. Further, <cit.>, implies that |λ(Ω)|≤ c_0√(|Ω|) for an absolute constant c_0 independent of Ω. In view of this it is natural to ask whether there exists a bounded, smooth domain Ω which maximises λ(Ω) among all other domains of the same volume. Physically this amounts to finding a domain Ω which supports a magnetic field (tangent to the boundary of Ω) whose field lines are (on average) as entangled as possible among all other magnetic fields and all other domains of the same volume.

Some necessary conditions, assuming the existence of optimal domains, have been derived in <cit.>. Up to this day the question of existence of optimal domains for this problem has remained open, even though some recent advances have been made within certain classes of domains, c.f. <cit.>,<cit.>.
Here we consider a corresponding problem for the surface helicity. The first result is the following.

	
	Let Σ⊂ℝ^3 be a closed, connected C^1,1-surface. Then there exists some v_±∈ L^2𝒱_0(Σ) with v_±_L^2(Σ)=1 and such that
	
    ±ℋ(v_±)=max_w≠ 0
    
    		w∈ L^2𝒱_0(Σ)±ℋ(w)/w^2_L^2(Σ)=:Λ_±(Σ).

	Further,
	
		
  * If Σ≅ S^2, then every non-zero element of L^2𝒱_0(Σ) realises the above Rayleigh quotients.
		
  * If Σ≇S^2, then v_±∈ℋ(Σ) (i.e. v_± are additionally curl-free).
	

To state the next result we denote by π the L^2-orthogonal projection from L^2𝒱(Σ) onto L^2𝒱_0(Σ).

	
	Let Σ⊂ℝ^3 be a closed, connected C^1,1-surface. Then the operator π∘BS_Σ:L^2𝒱_0(Σ)→ L^2𝒱_0(Σ) is compact and self-adjoint and admits a largest positive and smallest negative eigenvalue. Further, v∈ L^2𝒱_0(Σ) maximises helicity among all other fields w∈ L^2𝒱_0(Σ) of the same L^2-norm if and only if v is an eigenfield of π∘BS_Σ corresponding to the largest positive eigenvalue. A corresponding result holds for the helicity minimisation problem.


	In general, it is highly non-trivial to compute the corresponding eigenfields of π∘BS_Σ which realise the Rayleigh quotients in (<ref>) even for rotationally symmetric surfaces. However, we provide in <Ref> an, in comparison, easy way to compute these eigenfields as well as the corresponding values Λ_+(Σ), Λ_-(Σ) if the surface Σ is toroidal. This will allow us to obtain exact solutions in the case of rotationally symmetric surfaces.

We now deal with the question of optimising the following quantity, recall (<ref>),

    Λ(Σ):=max{Λ_+(Σ),Λ_-(Σ)}=max_w≠ 0
    
    			w∈ L^2𝒱_0(Σ)|ℋ(w)|/w^2_L^2(Σ)

among C^1,1-closed surfaces. Note that according to <Ref> we have Λ(Σ)≥ 0 with equality if and only if Σ≅ S^2. Hence, Σ is a global minimiser of Λ(Σ) if and only if Σ≅ S^2. Here, due to their relevance in plasma physics, we want to focus on surfaces Σ≅ T^2.

	
	Let T^2≅Σ⊂ℝ^3 be a closed, connected C^1,1-surface and let Ω be the domain of finite volume bounded by Σ. Then
	
    Λ(Σ)≥1/2.

	Equality holds if and only if ℋ(γ)=0, where γ is the L^2-orthogonal projection of Γ|_Σ onto ℋ(Σ) and Γ is any fixed non-zero harmonic Neumann field of Ω, i.e. any H^1-regular, curl-free, div-free vector field on Ω which is tangent to ∂Ω=Σ.


	With some more effort it should be possible to extend the same lower bound to surfaces of arbitrary (strictly positive) genus using similar methods as in the proof of <Ref> by working with a suitable basis of ℋ_N(Ω) in order to obtain an appropriate matrix representation of the surface Biot-Savart operator. However, we do not pursue such a possible extension further and instead list this as an open problem below.

Using the linking interpretation of helicity, <Ref> and <Ref>, we will be able to conclude the following.

	
	Let Σ⊂ℝ^3 be a closed, connected, C^1,1-regular, axisymmetric torus. Then Λ(Σ)=1/2. Consequently
	
    Λ(Σ)=min_T^2≅Σ∈ C^1,1Λ(Σ).



	
	We note that in <Ref> and <Ref> we did not specify the area of Σ. As we shall see, the scaling properties of the functional Λ imply that minimising or maximising Λ among surfaces with prescribed area is equivalent to optimising Λ among surfaces without an area constraint.

We recall that the corresponding 3-d problem motivated to try to find a global maximiser of Λ(Σ) rather than a global minimiser. We pose this as a remaining open problem.
Open Problems:

	
  * Does there exist a constant 0<c<∞ such that Λ(Σ)≤ c for all T^2≅Σ which are C^1,1-regular?
	
  * If the answer to (i) is positive, does there exist some T^2≅Σ such that Λ(Σ)=max_T^2≅Σ∈ C^1,1Λ(Σ)?
	
  * Do the answers to (i) and (ii) change if we drop the assumption T^2≅Σ and instead allow Σ to have arbitrary genus?
	
  * Does <Ref> extend to surfaces of arbitrary genus, i.e. is it true that every closed, connected C^1,1-surface Σ⊂ℝ^3 of genus g≥ 1 satisfies Λ(Σ)≥1/2?



 §.§ Simple Surface currents


When building plasma fusion confinement devices one wishes to generate magnetic fields in order to confine the fusion plasma. In particular, in so called stellarator devices, one uses a complex coil structure to generate the fields <cit.>. One way to model these complex coil structures is the so called coil winding surface (CWS) model, <cit.>, where one assumes that the coils are infinitely thin and wind around a given closed surface. The induced current is then modelled as a vector field tangent to the CWS and is assumed to be divergence-free in view of Maxwell's equations. Given a C^1,1-regular CWS Σ⊂ℝ^3 and a divergence-free current j on the CWS, the corresponding magnetic field induced by j within the domain Ω enclosed by Σ is given by the Biot-Savart law

    BS(j)(x)=1/4π∫_Σj(y)×x-y/|x-y|^3dσ(y), x∈Ω.

Now, for a fixed current j, there exist in general further currents j^' which induce the same magnetic field within Ω. In fact, since BS is a linear operator, we must have that j-j^'∈Ker(BS). To be more precise, we may view BS as a bounded operator from L^2𝒱_0(Σ) into L^2𝒱(Ω) (the square integrable vector fields on Ω), c.f. <cit.>. It has been further shown, see <cit.> that (Ker(BS))=g(Σ), where g(Σ) denotes the genus of Σ. In our applications, the CWSs are toroidal, i.e. T^2≅Σ, and thus (Ker(BS))=1. This gives us some flexibility and hence we may look for currents, and consequently coil configurations, which have a "simple" form. To this end we make the following definition.

	
	Let T^2≅Σ⊂ℝ^3 bound a C^1,1-solid torus Ω⊂ℝ^3. Given a purely poloidal curve σ_p and a toroidal curve σ_t we further let γ_t∈ℋ(Σ) denote the induced harmonic field. Given a current j∈ L^2𝒱_0(Σ) we define Q(j):=∫_Σj·γ_t dσ/|Σ| and say that j simple if Q(j)=0. 


	
	
		
  * In view of <Ref>, <Ref> coincides with <Ref> whenever j is of class C^0,1.
		
  * We have previously explained in <Ref> that γ_t is independent of the choice of σ_p and that changing σ_t may at most change γ_t to -γ_t and therefore the definition of a simple current is independent of the chosen curves σ_p and σ_t.
		
  * If j∈ C^0,1𝒱_0(Σ), then according to <Ref> we see that j is simple if and only if ∫_Σq̂(x)dσ(x)=0, or in other words if and only if, on average, the weighted asymptotic toroidal windings of the field lines of j are zero. Therefore, in an oversimplified manner of speaking, the field lines of a simple current should not form any "knotted" structures and be expected to tend to be either more poloidal or contractible in nature.
		
  * A word of caution: It might turn out that the field lines of j are all toroidal curves and that simply on average the corresponding field lines point as often in positive toroidal direction as they point in negative toroidal direction. As an example consider Σ⊂ℝ^3 to be a standard rotationally symmetric torus and let ϕ denote the toroidal coordinate and θ the poloidal coordinate. We may then define j:=sin(θ)e_ϕ, where e_ϕ denotes the standard normalised toroidal field. Then j∈ C^0,1𝒱_0(Σ) and Q(j)=0, as follows easily from <Ref>, and hence j is simple. However, j has only a toroidal component, see <Ref>.
	





Here is our main result regarding the existence of simple current configurations.

	
	Let T^2≅Σ⊂ℝ^3 bound a C^1,1-solid torus Ω⊂ℝ^3. Then for every j∈ L^2𝒱_0(Σ) there exists a unique j^'∈ L^2𝒱_0(Σ) such that BS(j^')=BS(j) in Ω and j^' is simple. Further, if j∈ L^p𝒱_0(Σ) for some 2≤ p≤∞ or j∈ C^0,α𝒱_0(Σ) for some 0≤α<1, then so is j^', where C^0,0≡ C^0 denotes the continuous currents.

In the context of plasma fusion confinement devices one of the situations one faces is the following: We suppose we are given a CWS Σ and a plasma region P which is assumed to be precompact within the finite domain Ω bounded by the CWS and is thought of to contain the fusion plasma. Further, we assume we are given a square integrable, harmonic, target magnetic field B_T∈ L^2ℋ(P):={X∈ L^2(P,ℝ^3)|curl(X)=0=div(X)} which we wish to generate by a surface current in order to confine the fusion plasma. It is known, see <cit.>, that under reasonable assumptions, the image of BS, with domain L^2𝒱_0(Σ), is contained in and L^2(P)-dense in the space L^2ℋ(P). We therefore obtain the following important corollary from <Ref>, where the slightly technical assumptions in the upcoming results are depicted in <Ref>.





	
	Let T^2≅Σ⊂ℝ^3 bound a C^1,1-solid torus Ω⊂ℝ^3. Assume further, that we are given another C^1,1-solid torus P which is contained in Ω and of positive distance to Σ and that there exists a C^2-disc D⊂Ω such that ∂ D⊂Σ is a purely poloidal C^1-curve in Σ and such that D∩ P is also a disc bounded by a purely poloidal C^1-curve contained in ∂ P. Then for every B_T∈ L^2ℋ(P) and every ϵ>0 there exists a current j∈ L^2𝒱_0(Σ) with Q(j)=0 and BS(j)-B_T_L^2(P)≤ϵ.


Our final result provides a minimisation procedure which allows one to obtain the desired approximating simple currents as minimisers of a specific "energy functional". The procedure is a modification  of a minimisation procedure which has already been used successfully in the past. We first recall the already established procedure, c.f. <cit.>. Given a positive parameter λ>0 one can consider the following minimisation problem

    C(λ;B_T):=inf_j∈ L^2𝒱_0(Σ)(BS(j)-B_T^2_L^2(P)+λj^2_L^2(Σ)).

It follows from standard variational techniques and the convexity of the functional involved that the infimum is in fact a minimum and that there exists a unique current j_λ∈ L^2𝒱_0(Σ) which realises C(λ;B_T). Using the density of the image of the Biot-Savart operator within L^2ℋ(P) it has been shown that BS(j_λ)-B_T_L^2(P)→ 0 as λ↘ 0, see <cit.>. In general, there is no reason to expect that the obtained currents j_λ satisfy Q(j_λ)=0. However, one can enforce this condition by observing that 

    L^2𝒱^Q=0_0(Σ):={j∈ L^2𝒱_0(Σ)|Q(j)=0}

is an L^2-closed subspace, and hence a Hilbert space in its own right, and by setting up a corresponding minimisation procedure

    C_0(λ;B_T):=inf_j∈ L^2𝒱^Q=0_0(Σ)(BS(j)-B_T^2_L^2(P)+λj^2_L^2(Σ)).

Just like in the case of C(λ;B_T) it follows from standard arguments that the infimum in C_0(λ;B_T) is actually achieved and that the global minimiser is unique. Our final result is the following

	
	Let T^2≅Σ⊂ℝ^3 bound a C^1,1-solid torus Ω⊂ℝ^3. Assume further that we are given another C^1,1-solid torus P which is contained in Ω and of positive distance to Σ and that there exists a C^2-disc D⊂Ω such that ∂ D⊂Σ is a purely poloidal C^1-curve in Σ and such that D∩ P is also a disc bounded by a purely poloidal C^1-curve contained in ∂ P. Given any B_T∈ L^2ℋ(P) we denote by j^0_λ the (unique) global minimiser realising C_0(λ;B_T) in (<ref>) for fixed λ>0. Then BS(j^0_λ)-B_T_L^2(P)→ 0 as λ↘ 0.



§ PROOFS OF MAIN RESULTS




 §.§ Proof of <Ref>

Before we come to the proof let us recall how the (0,2,3)-helicity is defined. Given a closed 1-form α on a surface Σ we can consider the inclusion ι: (Σ×Σ)∖{x=y}→Σ×Σ and the projections π_x:Σ×Σ→Σ,(x,y)↦ x and π_y:Σ×Σ→Σ,(x,y)↦ y. We define the Gauss map g:(Σ×Σ)∖{x=y}→ S^2, (x,y)↦x-y/|x-y| and we let ω^2_S^2 denote the standard normalised(!) area form on S^2. We can then define α_x:=(π_x)^#α, α_y:=(π_y)^#α (here f^# denotes the pullback via a map f). Now, strictly speaking, the space C_2(Σ):=(Σ×Σ)∖{x=y} is first compactified by means of the Fulton-MacPherson compactification, c.f. <cit.>, and it is shown that the forms α_x, α_y as well as the Gauss map extend smoothly to the compactification <cit.>. Further, the compactification has a manifold structure <cit.>. Denoting by C_2[Σ] the compactification and with a slight abuse of notation we can define the helicity as

    ℋ_(0,2,3)(α):=∫_C_2[Σ]α_x∧α_y∧ g^#ω^2_S^2.

We note that the space C_2[Σ] is 4-dimensional and so it makes sense to integrate a 4-form. The same reasoning as in <cit.> tells us that it is enough to prove that for all (x,y)∈ (Σ×Σ)∖{x≠ y} we have the identity

    α_x∧α_y∧ g^#ω^2_S^2=1/4πv(x)·(v(y)×x-y/|x-y|^3)ω_Σ(x)∧ω_Σ(y)

where ω_Σ is the area-form on Σ induced by the Euclidean metric and where v is the unique vector field on Σ satisfying ι_vω_Σ=α.

	We prove more generally that if α and β are two 1-forms on Σ, then
	
    β_x∧α_y∧ g^#ω^2_S^2=1/4πw(x)·(v(y)×x-y/|x-y|^3)ω_Σ(x)∧ω_Σ(y)

	where w is the vector field associated with β and v is the vector field associated with α by means of contracting the area form. We fix any x,y∈Σ with x≠ y and let μ,η be charts of Σ around x and y respectively. Additionally, we choose μ×η as a chart around (x,y)∈ (Σ×Σ)∖{x=y}. Further, we let p,q be any two points contained in the domains of μ and η respectively. We see that, if we denote by dμ^i,dη^j the induced covector bases at p,q∈Σ respectively and with a slight abuse of notation by dμ^i(p,q),dη^j(p,q) the induced basis at (p,q)∈ (Σ×Σ)∖{x=y}, then, if we express β(p)=β_i(p)dμ^i(p), we find 
	
    β_x(p,q)=((π_x)^#β)(p)=∑_i=1^2β_i(p)dμ^i(p,q).

	Similarly, α_y(p,q)=∑_j=1^2α_j(q)dη^j(p,q). As for the remaining term g^#ω^2_S^2 we recall that we have the identity
	
    ω^2_S^2=1/4πι^#(x dy∧ dz-ydx∧ dz+zdx∧ dy)

	where ι denotes the inclusion S^2↪ℝ^3. Hence, we may view the Gauss map g as a map into ℝ^3 rather than S^2 and are left with computing the pullback
	
    G^#(x dy∧ dz-ydx∧ dz+zdx∧ dy)(p,q)

	where G:(Σ×Σ)∖{x=y}→ℝ^3,(x,y)↦x-y/|x-y|. In order to find an appropriate expression for the coefficients of this 2-form we observe that
	
    G_*(p,q)∂_μ^i(p,q)=∂_μ^i(p)+(p-q)·∂_μ^i(p)/|p-q|^2(q-p)/|p-q|
    
    		G_*(p,q)∂_η^j(p,q)=-∂_η^j(q)+(p-q)·∂_η^j(q)/|p-q|^2(q-p)/|p-q|

	where the vectors tangent to Σ are viewed as vectors in ℝ^3. Defining ω(x,y,z):=xdy∧ dz-ydx∧ dz+z dx∧ dy, we observe that by definition of the pullback we have
	
    (G^#ω)(p,q)(∂_μ^1(p,q),∂_μ^2(p,q))=ω(p-q/|p-q|)(G_*(p,q)∂_μ^1(p,q),G_*(p,q)∂_μ^2(p,q)).

	Letting r̂:=p-q/|p-q|, ν:=∂_μ_1(p) and ξ:=∂_μ_2(p) we note that we need therefore to compute
	
    ω(r̂)(ν-(r̂·ν)r̂,ξ-(r̂·ξ)r̂)

	and an explicit computation yields
	
    ω(r̂)(ν-(r̂·ν)r̂,ξ-(r̂·ξ)r̂)=(ν×ξ)·r̂.

	Using this formula we can compute all the coefficient functions and eventually arrive at
	
    (G^#ω)(p,q)
    		
    
    		=p-q/4π|p-q|^3·(∂_μ^1(p)×∂_μ^2(p)dμ^1(p,q)∧ dμ^2(p,q)+∂_η^1(q)×∂_η^2(q)dη^1(p,q)∧ dη^2(p,q).
    		
    . -∑_i,j=1^2∂_μ^i(p)×∂_η^j(q)dμ^i(p,q)∧ dη^j(p,q))

	where we again view the tangent vectors at Σ as vectors in ℝ^3. We observe that the 2-form β_x∧α_y only contains basis vectors of the form dμ^i(p,q)∧ dη^j(p,q) so that all the dμ^1∧ dμ^2 and dη^1∧ dη^2 terms from G^#ω will vanish in the product β_x∧α_y∧ G^#ω because at least one of the covectors dμ^i or dη^j in these expressions must repeat, so that only the mixed terms are of relevance. We obtain
	
    β_x(p,q)∧α_y(p,q)∧ g^#ω^2_S^2(p,q)
    		
    
    		=q-p/4π|p-q|^3·∑_i,j,k,m=1^2β_k(p)α_m(p)(∂_μ^i(p)×∂_η^j(q))dμ^k(p,q)∧ dη^m(p,q)∧ dμ^i(p,q)∧ dη^j(p,q)
    		
    
    		=q-p/4π|p-q|^3·((β_2(p)∂_μ_1(p)-β_1(p)∂_μ_2(p))× (α_1(q)∂_η^2(q)-α_2(q)∂_η^1(q)))dμ^1∧ dμ^2∧ dη^1∧ dη^2.

	We recall that we started with two points x,y∈Σ with x≠ y and μ and η were any arbitrary charts centred around these two points. In particular, we may pick our charts such that g_ij(x)=δ_ij=g̃_ij(y) at these two points where g_ij and g̃_ij denote the metric tensor coefficients in the respective coordinate charts. Further, with the right choice of orientation on Σ we may assume that ∂_μ^1(x)×∂_μ^2(x)=𝒩(x) where 𝒩 denotes the outward pointing unit normal. Using the musical isomorphism we can then identify the forms α and β with vector fields X and Z respectively (note that this identification differs from identifying a 1-form by contracting the area-form). We can then expand (x-y) in the basis ∂_μ^1(x),∂_μ^2(x),𝒩(x) and using the properties of the chart we find
	
    (x-y)× (β_2(x)∂_μ^1(x)-β_1(x)∂_μ^2(x))=[(y-x)· Z(x)]𝒩(x)+[(x-y)·𝒩(x)]Z(x).

	Applying the cyclic properties of the inner product we find
	
    (x-y)· [(β_2(x)∂_μ^1(x)-β_1(x)∂_μ^2(x))× (α_1(y)∂_η^2(y)-α_2(y)∂_η^1(y))]
    		
    
    		=(α_1(y)∂_η^2(y)-α_2(y)∂_η^1(y))·([(y-x)· Z(x)]𝒩(x)+[(x-y)·𝒩(x)]Z(x)).

	Now we note that if a∈ℝ^3 is an arbitrary vector, then upon expanding it in the (orthonormal) basis {∂_η^1(y),∂_η^2(y),𝒩(y)}, we obtain
	
    𝒩(y)· [X(y)× a]=α_1(y)a^2-α_2(y)a^1=(α_1(y)∂_η^2(y)-α_2(y)∂_η^1(y))· a.

	We hence arrive at
	
    β_x(x,y)∧α_y(x,y)∧ g^#ω^2_S^2(x,y)
    		
    
    		=𝒩(y)/4π|x-y|^3·[[((y-x)· Z(x))𝒩(x)+((x-y)·𝒩(x))Z(x)]× X(y)]ω_Σ(x)∧ω_Σ(y)
    		
    
    		=X(y)×𝒩(y)/4π|x-y|^3· [((y-x)· Z(x))𝒩(x)+((x-y)·𝒩(x))Z(x)]ω_Σ(x)∧ω_Σ(y)
    		
    
    		=X(y)×𝒩(y)/4π|x-y|^3· [(Z(x)×𝒩(x))× (y-x)]ω_Σ(x)∧ω_Σ(y)

	where we used the vector-triple product rule in the last step. Lastly, we observe that the vector field associated with a 1-form α by contracting the area-form can be obtained from the vector field associated with the 1-form α via the musical isomorphism by taking the cross product with the outward pointing unit normal. So if we let v,w be the vector fields associated with the 1-forms α and β by contracting the area-form respectively, we arrive at
	
    β_x(x,y)∧α_y(x,y)∧ (g^#ω^2_S^2)(x,y)=v(y)·(w(x)×y-x/4π|x-y|^3)ω_Σ(x)∧ω_Σ(y).

	Consequently
	
    ∫_C_2[Σ]β_x∧α_y∧ g^#ω^2_S^2=∫_Σ v(y)·BS_Σ(w)(y)dσ(y)

	and hence ℋ_(0,2,3)(α)=ℋ(v).



 §.§ Proof of <Ref>

We recall that we want to prove that for a given closed, connected surface Σ⊂ℝ^3 there exists some v∈ L^2𝒱_0(Σ) with ℋ(v)≠ 0 if and only if g(Σ)≥ 1, where g(Σ) denotes the genus of Σ. In addition, we also need to show that ℋ_c(v,w)=0 for all w∈ L^2𝒱_0(Σ) whenever v is a co-exact vector field, where ℋ_c denotes the cross-helicity and ℋ denotes the helicity of vector fields.

	
	Step 1: We first prove that if g(Σ)≥ 1, then there exists some element v∈ L^2𝒱_0(Σ) with ℋ(v)≠ 0.
	We first observe that each closed surface Σ bounds a bounded domain Ω⊂ℝ^3 with ∂Ω=Σ. We then have the well-known relation (ℋ_N(Ω))=g(Σ), where ℋ_N(Ω) denotes the space of curl-free, div-free, H^1-vector fields on Ω which are tangent to the boundary ∂Ω and g(Σ) denotes the genus of Σ, <cit.>. In our situation g(Σ)≥ 1 and so we can fix some non-zero Γ∈ℋ_N(Ω). Since Γ is tangent to Σ we may view it as a vector field on Σ. The fact that Γ is curl-free on Ω implies that so is its restriction to Σ (in the weak sense). We recall that ℋ_N(Ω)⊂⋂_0<α<1C^0,α𝒱(Ω) and hence we can write according to the Hodge decomposition theorem Γ|_Σ=grad_Σ(f)+γ for suitable f∈⋂_0<α<1C^1,α(Σ) and γ∈ℋ(Σ) where ℋ(Σ) denotes the space of curl-free and div-free fields on Σ of class ⋂_1≤ p<∞W^1,p. Since Γ is non-zero so must be γ (since if the restriction of a harmonic Neumann field to the boundary is a gradient field, then the harmonic Neumann field is already identically zero). We can now let 𝒩 denote the outward pointing unit normal and define γ̃:=γ×𝒩 (so that in turn γ=𝒩×γ̃). This new field γ̃ is still divergence-free and curl-free (in the language of differential forms the 1-form associated with γ̃ by the musical isomorphism can be obtained from that of γ by applying the Hodge-star operator). If either ℋ(γ)≠ 0 or ℋ(γ̃)≠ 0, then the theorem is proven (since both vector fields are div-free). If on the other hand ℋ(γ)=0=ℋ(γ̃), then we can consider the vector field v:=γ̃+γ and notice that, due to the symmetry property of the Biot-Savart operator, we have
	
    ℋ(v)=ℋ(γ)+ℋ(γ̃)+2ℋ_c(γ̃,γ)=2ℋ_c(γ̃,γ)

	where we recall that ℋ_c(γ̃,γ)=(γ̃,BS_Σ(γ))_L^2(Σ) is the "cross-helicity". We note that since  γ̃=γ×𝒩 we get by means of the cyclic properties of the Euclidean product
	
    4πℋ_c(γ̃,γ)=∫_Σ∫_Σ(γ̃(x)×γ(y))·x-y/|x-y|^3dσ(x)dσ(y)
    		
    
    		=∫_Σ∫_Σ((γ(x)×𝒩(x))×γ(y))·x-y/|x-y|^3dσ(x)dσ(y)
    		
    
    		=∫_Σ∫_Σ [(γ(x)·γ(y))𝒩(x)-(γ(y)·𝒩(x))γ(x)]·x-y/|x-y|^3dσ(y)dσ(x)
    		
    
    		=∫_Σ∫_Σγ(y)·[(𝒩(x)·x-y/|x-y|^3)γ(x)-(γ(x)·x-y/|x-y|^3)𝒩(x)]dσ(y)dσ(x).

	We first claim that
	
    ℝ^3→ℝ, y↦∫_Σ(γ(x)·x-y/|x-y|^3)(𝒩(x)-𝒩̃(y))dσ(x)

	is a continuous function, where 𝒩̃ is any C^0,1-regular (compactly supported) extension of 𝒩 to ℝ^3. But this follows easily from the generalised dominated convergence theorem because we have the estimate
	
    |(γ(x)·x-y/|x-y|^3)(𝒩(x)-𝒩̃(y))|≤c/|x-y|

	for a suitable constant c>0 which is independent of y and from the fact that the function ℝ^3→ℝ, y↦∫_Σ1/|x-y|dσ(x) is continuous.
	
	Therefore, if for some fixed y∈Σ we let (y_n)_n⊂Ω be any sequence converging to y, we find
	
    γ(y)·∫_Σ(γ(x)·x-y/|x-y|^3)𝒩(x)dσ(x)
    		
    
    		=γ(y)·∫_Σ(γ(x)·x-y/|x-y|^3)(𝒩(x)-𝒩(y))dσ(x)
    		
    
    		=lim_n→∞γ(y)·∫_Σ(γ(x)·x-y_n/|x-y_n|^3)(𝒩(x)-𝒩̃(y_n))dσ(x)

	where we used that γ(y)·𝒩(y)=0 because γ is a tangent field of Σ. We observe that since y_n∉Σ and since 𝒩̃(y_n) is independent of x, the term involving 𝒩̃(y_n) vanishes because
	
    ∫_Σγ(x)·x-y_n/|x-y_n|^3dσ(x)=-∫_Σγ(x)·∇^Σ_x(1/|x-y_n|)dσ(x)=0 for all n

	where we used that γ is divergence-free and that x↦1/|x-y_n| defines a C^1-function on Σ for all fixed n because each y_n has a positive distance to Σ. We conclude
	
    γ(y)·∫_Σ(γ(x)·x-y/|x-y|^3)𝒩(x)dσ(x)=lim_n→∞γ(y)·∫_Σ(γ(x)·x-y_n/|x-y_n|^3)𝒩(x)dσ(x).

	On the other hand we have the well-known jump formula, <cit.> which tells us that we can find a suitable sequence (y_n)_n⊂Ω converging to y such that
	
    ∫_Σ(𝒩(x)·x-y/|x-y|^3)γ(x)dσ(x)=-4π/2γ(y)+lim_n→∞∫_Σ(𝒩(x)·x-y_n/|x-y_n|^3)γ(x)dσ(x).

	Combining our considerations so far, we see that for all y∈Σ there exists some sequence (y_n)_n⊂Ω converging to y such that
	
    4πℋ_c(γ,γ̃)=-4π/2γ^2_L^2(Σ)
    
    		+∫_Σlim_n→∞γ(y)·(∫_Σ(𝒩(x)·x-y_n/|x-y_n|^3)γ(x)dσ(x)-∫_Σ(γ(x)·x-y_n/|x-y_n|^3)𝒩(x)dσ(x))dσ(y).

	We observe now that γ admits a curl- and div-free extension to Ω, i.e. there exists some ṽ∈⋂_1≤ p<∞W^1,p𝒱(Ω) with curl(ṽ)=0, div(ṽ)=0 in Ω and ṽ^∥=γ, where ṽ^∥:=ṽ-(𝒩·ṽ)𝒩. Indeed, we recall that Γ|_Σ=grad_Σ(f)+γ for a suitable function f∈⋂_0<α<1C^1,α(Σ), and thus if we let f̃ denote the harmonic extension of f, c.f. <cit.>, then Γ-grad(f̃)∈⋂_1≤ p<∞W^1,p𝒱(Ω) is the desired curl- and div-free extension. One can then use Gauss' theorem and the fact that y_n∈Ω for fixed n, to obtain
	
    ∫_Σ(𝒩(x)·x-y_n/|x-y_n|^3)γ(x)dσ(x)-∫_Σ(γ(x)·x-y_n/|x-y_n|^3)𝒩(x)dσ(x)
    		
    
    		=∫_Σ(𝒩(x)·x-y_n/|x-y_n|^3)ṽ(x)dσ(x)-∫_Σ(ṽ(x)·x-y_n/|x-y_n|^3)𝒩(x)dσ(x)
    		
    
    		=4πṽ(y_n)+∫_Ωdiv(ṽ)(x)x-y_n/|x-y_n|^3d^3x+∫_Ωy_n-x/|y_n-x|^3×curl(ṽ)(x)d^3x=4πṽ(y_n),

	where the first identity follows from the fact that the terms involving the normal part of ṽ cancel each other. Taking the limit n→∞ we conclude
	
    4πℋ_c(γ,γ̃)=4π/2γ^2_L^2(Σ).

	We recall that ℋ(γ+γ̃)=2ℋ_c(γ,γ̃) and therefore we arrive at
	
    ℋ(γ+γ̃)=2ℋ_c(γ,γ̃)=γ^2_L^2(Σ)>0

	because γ≢0. Hence, in any case, there must exist some γ̂∈ℋ(Σ) satisfying ℋ(γ̂)≠ 0.
	Step 2: In this step we prove that if w∈ L^2𝒱_0(Σ) and f∈ H^1(Σ), then ℋ_c(∇^⊥ f,w)=0, where ∇^⊥ f:=∇_Σf×𝒩. In particular, as ℋ(∇^⊥ f)=ℋ_c(∇^⊥ f,∇^⊥ f) we will get ℋ(∇^⊥ f)=0. Since on a genus zero surface all divergence-free fields are co-exact this will conclude the proof.
	First we observe that by a density argument we may without loss of generality assume that w,∇_Σf∈⋂_1≤ p<∞W^1,p𝒱(Σ). We follow the arguments of the first step. We simply replace γ(y) by w(y) and γ(x) by ∇_Σ f(x) in the arguments. The only step where the arguments need to be modified is in (<ref>). More specifically, the argument that γ(y)·𝒩̃(y_n) ∫_Σ(γ(x)·x-y_n/|x-y_n|^3)dσ(x) vanishes no longer applies because in our new situation γ(x) has to be replaced by ∇_Σ f(x) which is no longer divergence-free. However, we see that
	
    ∫_Σ(∇_Σf(x)·x-y_n/|x-y_n|^3)dσ(x)=-∫_Σ∇_Σ f(x)·∇^Σ_x1/|x-y_n|dσ(x)=∫_ΣΔ_Σf(x)/|x-y_n|dσ(x)

	where Δ_Σ:=div_Σ∘∇_Σ denotes the Laplace operator on Σ. We note that the function ℝ^3→ℝ, y↦∫_Σh(x)/|x-y|dσ(x) is continuous for any function h∈ L^p(Σ), for some p>2, and therefore
	
    lim_n→∞w(y)·𝒩̃(y_n)∫_Σ(∇_Σ f(x)·x-y_n/|x-y_n|^3)dσ(x)=w(y)·𝒩(y)∫_ΣΔ_Σ f(x)/|x-y|dσ(x)=0

	because 𝒩̃(y)=𝒩(y) (since 𝒩̃ is an extension of 𝒩) and because w(y) is tangent to Σ. The remaining arguments apply verbatim which leads us to
	
    2ℋ_c(∇^⊥ f,w)=( ∇_Σ f,w)_L^2(Σ)=0

	because w is divergence-free.



 §.§ Proof of <Ref>


	We recall that v∈ C^0,1𝒱_0(Σ) is a div-free Lipschitz vector field on Σ. Further, we denote by γ_x the (unique) integral curve of v starting at x and for given T>0 we denote by γ_x[0,T] the path along the field line γ_x from time t=0 until t=T.
	Step 1: In the first step we prove that for every T,S∈ (0,∞) the two path γ_x[0,T] and γ_y[0,S] are disjoint almost surely (and hence by construction the artificially closed path σ^τ_x,T and σ^-τ_y,S will be also almost surely disjoint and have a well-defined linking number).
	
	For given T,S>0 we consider the set B_T,S:={(x,y)∈Σ×Σ|γ_x[0,T]∩γ_y[0,S]≠∅}. Now we note that (x,y)∈ B_T,S⇔∃ z∈γ_x[0,T]∩γ_y[0,S].	
	
	In conclusion we must have γ_x(τ)=z=γ_y(λ)⇔ y=γ_x(τ-λ) for suitable 0≤τ≤ T, 0≤λ≤ S, where we used the properties of the flow. Hence, for any T,S>0 we have
	
    B_T,S⊂⋃_n∈ℕ{(x,y)∈Σ×Σ|y∈γ_x(-n,n)}.

	We observe that the right hand side is independent of T and S and that by sigma-subadditivity it is enough to prove that {(x,y)∈Σ×Σ|y∈γ_x(-n,n)} is a null-set in order to show that for almost every (x,y)∈Σ×Σ the sets γ_x[0,T] and γ_y[0,S] do not intersect (in fact this null set is independent of the choice of T and S).
	
	Denoting by μ the Riemannian measure on Σ we find, using Fubini's theorem,
	
    (μ×μ)({(x,y)∈Σ×Σ|y∈γ_x(-n,n)})=∫_Σ∫_Σχ_{(x,y)∈Σ×Σ|y∈γ_x(-n,n)}dσ(y)dσ(x).

	We observe that
	
    χ_{(x,y)∈Σ×Σ|y∈γ_x(-n,n)}(x,y)=χ_B_x(y)

	where B_x:={y∈Σ|y∈γ_x(-n,n)} so that
	
    ∫_Σχ_B_x(y)dσ(y)=μ(B_x)=μ(γ_x(-n,n))=0

	where we used in the last step that γ_x(-n,n) is either a point or otherwise a local embedding of a 1-manifold and hence has Hausdorff-dimension at most 1.
	Step 2: In this step we prove the remaining claims of the theorem.
	In order to specify our choice of τ(Σ)>0, we observe first that there exists some c=c(Σ)>0 such that, c.f. <cit.>,
		
    |𝒩(x)· (x-y)|≤ c|x-y|^2 for all x,y∈Σ.

		Further, we note that the function
		
    κ:Σ×Σ→ℝ, (x,y)↦𝒩(x)·𝒩(y)

		is uniformly continuous and thus
		
    𝒩(x)·𝒩(y)≥1/2 for all x,y∈Σ, |x-y|<δ(Σ)

		for a suitable δ(Σ)>0. We can then choose τ(Σ)>0 such that τ(Σ)≤min{δ/3,1/6c} and such that Ψ_t(x):=x+t𝒩(x) is a diffeomorphism from Σ onto its image for every |t|≤τ.
		
		In order to obtain the desired linking interpretation we observe first that, c.f. <cit.>:
		
		For every f∈ L^1(Σ×Σ), the sequence of functions ∫_0^S_n∫_0^T_nf(γ_x(t),γ_y(s))dtds/T_nS_n admits a well-defined L^1-limit denoted by f̂∈ L^1(Σ×Σ) and 
    ∫_Σ×Σf(x,y)dσ(x)dσ(y)=∫_Σ×Σf̂(x,y)dσ(x)dσ(y).

		We define f(x,y):=1/4π(v(x)× v(y))·x-y/|x-y|^3∈ L^1(Σ×Σ), which can be seen by writing v(x)× v(y)=v(x)× (v(y)-v(x)), using the Lipschitz property of v and the fact that 1/|x-y| is integrable over Σ×Σ. We therefore arrive at
		
    ℋ(v)=1/4π∫_Σ×Σlim^L^1_n→∞∫_0^T_n∫_0^S_n(γ̇_x(t)×γ̇_y(s))·γ_x(t)-γ_y(s)/|γ_x(t)-γ_y(s)|^3dtds/T_nS_ndσ(x)dσ(y).

		We recall the linking integral for two piecewise differentiable closed curves σ_1,σ_2 with σ_i(0)=σ_i(T_i)
		
    lk(σ_1,σ_2)=1/4π∫_0^T_1∫_0^T_2(σ̇_1(t)×σ̇_2(s))·σ_1(t)-σ_2(s)/|σ_1(t)-σ_2(s)|^3dtds.

		Now, as discussed before the statement of <Ref>, we close the curves γ_x([0,T_n]) and γ_y([0,S_n]) and obtain closed curves σ^τ_x,T_n and σ^-τ_y,S_n. For notational simplicity we drop the index n in T_n and S_n. According to (<ref>) and (<ref>) we have to show that the additional terms involved in the expression of lk(σ^τ_x,T,σ^-τ_y,S) converge to zero in the L^1(Σ×Σ)-sense.
		
		To this end we first let a∈Σ∖γ_x(ℝ), 0≤ s≤τ be any fixed elements and we observe that
		
    |a-s𝒩(a)-γ_x(t)|^2=|a-γ_x(t)|^2+s^2-2s(a-γ_x(t))·𝒩(a).

		Now, since a,γ_x(t)∈Σ we can use the estimate |(a-γ_x(t))·𝒩(a)|≤ c|a-γ_x(t)|^2 so that
		
    |a-s𝒩(a)-γ_x(t)|^2≥ |a-γ_x(t)|^2+s^2-2cs|a-γ_x(t)|^2
    			
    ≥ |a-γ_x(t)|^2+s^2/2-2c^2|a-γ_x(t)|^4

		where we used the elementary inequality 2db≤d^2/ϵ+ϵ b^2 for all d,b∈ℝ, ϵ>0 with d=s, b=c|a-γ_x(t)|^2 and ϵ=2. Now we distinguish two cases: Either 2c^2|a-γ_x(t)|^2≤1/2⇔ |a-γ_x(t)|^2≤1/4c^2, in which case we obtain
		
    |a-s𝒩(a)-γ_x(t)|^2≥|a-γ_x(t)|^2+s^2/2≥|a-γ_x(t)|^2+s^2/4,

		otherwise we find |a-γ_x(t)|^2≥1/4c^2. But in this case we can use the estimate |2s(a-γ_x(t))·𝒩(a)|≤ 2s^2+|a-γ_x(t)|^2/2 and therefore in this case
		
    |a-s𝒩(a)-γ_x(t)|^2≥|a-γ_x(t)|^2/2-s^2≥|a-γ_x(t)|^2/4+1/16c^2-s^2.

		To estimate the last term we recall that we are interested only in the situation 0≤ s≤τ and thus we can use the fact that τ≤1/6c, i.e. 1/36c^2≥τ^2 and therefore 1/16c^2≥ 2τ^2≥ 2s^2 which yields
		
    |a-s𝒩(a)-γ_x(t)|^2≥|a-γ_x(t)|^2+s^2/4

		in this case. We conclude that in any case we can estimate a part of the linking integral as follows:
		
    |∫_0^T∫_0^τ(v(γ_x(t))×𝒩(a))·a-s𝒩(a)-γ_x(t)/|a-s𝒩(a)-γ_x(t)|^3dsdt|≤ 4c_v∫_0^T∫_0^τ1/|a-γ_x(t)|^2+s^2dsdt

		for a suitable constant c_v>0 which is given by the (finite) C^0(Σ) norm of v. We can further explicitly compute and estimate
		
    ∫_0^τ1/|a-γ_x(t)|^2+s^2ds=arctan(τ/|a-γ_x(t)|)/|a-γ_x(t)|≤π/2|a-γ_x(t)|.

		We recall that in our application we want to connect either the point y or γ_y(S) with the corresponding points on Σ_-τ via a straight line following the normal direction of the starting point. We note that γ_y(S)=ψ_S(y) where ψ denotes the flow of v and because v is divergence-free, the corresponding flow is area-preserving. Hence we want to consider the situation where a=ψ(y) for a suitable area preserving diffeomorphism of Σ. But then we can estimate
		
    ∫_Σ×Σ|∫_0^T∫_0^τ(v(γ_x(t))×𝒩(ψ(y)))·ψ(y)-s𝒩(ψ(y))-γ_x(t)/|ψ(y)-s𝒩(ψ(y))-γ_x(t)|^3dsdt|dσ(x)dσ(y)
    			
    ≤ 2π c_v∫_Σ×Σ∫_0^T1/|ψ(y)-γ_x(t)|dtdσ(x)dσ(y)=2π c_vT∫_Σ×Σ1/|x-y|dσ(x)dσ(y)

		where we used Fubini's theorem in the last step and the fact that ψ as well as γ_x(t)=ψ_t(x) are area preserving diffeomorphisms for every t. Therefore, if we divide the above expression by T and S and take the limit T,S→∞ we see that the corresponding part of the linking integral converges to zero in the L^1-sense.
		
		Now we connect Ψ_-τ(y) and Ψ_-τ(γ_y(S)) on the surface Σ_-τ by a curve γ which is of unit speed and whose length is uniformly bounded independently of y and S, for instance we can connect them by a length minimising geodesic on Σ_-τ in which case its length is always bounded by the intrinsic diameter of Σ_-τ. In that case we can use the fact that Σ_-τ has a positive distance to Σ and therefore |γ_x(t)-γ(s)|≥ d(-τ)>0 for a suitable d which is independent of x,t,y,S and s. Consequently one can use the rough, pointwise, upper bound
		
    |v(γ_x(t))×γ̇(s)·γ_x(t)-γ(s)/|γ_x(t)-γ(s)|^3|≤c_v/|γ_x(t)-γ(s)|^2≤c_v/d^2

		in order to see that the corresponding part of the linking integral converges to zero in the L^1-sense.
		The only remaining part of the linking integral which is not covered by the considerations made so far is of the following type
		
    ∫_0^τ∫_0^τ𝒩(b)×𝒩(a)·b+t𝒩(b)-a+s𝒩(a)/|b+t𝒩(b)-a+s𝒩(a)|^3dsdt

		where in our application a∈{y,γ_y(S)} and b∈{x,γ_x(T)}. But here we can argue similar in spirit as in the first case, namely we can consider
		
    |a-s𝒩(a)-(b+t𝒩(b))|^2=|(b-a)+t𝒩(b)+s𝒩(a)|^2
    			
    
    			=|b-a|^2+|s𝒩(a)+t𝒩(b)|^2+2s𝒩(a)· (b-a)+2t𝒩(b)· (b-a).

		We note that, since a,b∈Σ and 0≤ t≤τ, |2t𝒩(b)· (b-a)|≤ 2cτ |b-a|^2. We recall that τ≤1/6c and thus |2t𝒩(b)· (b-a)|≤|b-a|^2/3 and so we arrive at
		
    |a-s𝒩(a)-(b+t𝒩(b))|^2≥|b-a|^2/3+|s𝒩(a)+t𝒩(b)|^2=|b-a|^2/3+t^2+s^2+2st𝒩(a)·𝒩(b).

		We recall that if |a-b|≤δ(Σ) then 𝒩(a)·𝒩(b)≥1/2 and hence, since s,t≥ 0 we arrive in this case at
		
    |a-s𝒩(a)-(b+t𝒩(b))|^2≥|a-b|^2/3+t^2+s^2≥|a-b|^2+t^2+s^2/6≥|a-b|^2+t^2/6.

		On the other hand, if |a-b|≥δ, we can estimate 2st≤t^2/2+2s^2 and thus
		
    |a-s𝒩(a)-(b+t𝒩(b))|^2≥|b-a|^2/3+t^2/2-s^2.

		Further, |b-a|^2/6≥δ^2/6≥τ^2≥ s^2 for all 0≤ s≤τ because by choice of τ we have τ≤δ/3. Consequently we arrive at
		
    |a-s𝒩(a)-(b+t𝒩(b))|^2≥|b-a|^2+t^2/6 for all a,b∈Σ.

		From here on out we can argue identically as in the first part of the proof that the corresponding part of the linking integral converges to zero in the L^1-sense. Hence, combining our findings with (<ref>) and the definition of the linking integral (<ref>) we conclude the validity of the theorem.



 §.§ Proof of <Ref>

We start by proving <Ref>.

	We will prove the statement only for Q(v), since an identical reasoning applies to P(v). We recall first that for given v∈ C^0,1𝒱_0(Σ) we denote its integral curves starting at a point x∈Σ by γ_x. Further we have by definition q̂(x)=lim_T→∞1/T∫_γ_x[0,T]γ_t where γ_t∈ℋ(Σ) is a harmonic field on Σ defined by the relations ∫_σ_pγ_t=0 and ∫_σ_tγ_t=1, where σ_p and σ_t are some fixed purely poloidal and a toroidal closed curves respectively. Finally, if q̂∈ L^1(Σ), we had set Q(v):=∫_Σq̂(x)dσ(x)/|Σ|.
	To see that q̂∈ L^1(Σ) and Q(v)=∫_Σv(x)·γ_t(x)dσ(x)/|Σ| we spell out the definition of the line integral, which leads us to
	
    ∫_γ_x[0,T]γ_t=∫_0^T γ̇_x(τ)·γ_t(γ_x(τ))dτ=∫_0^Tv(γ_x(τ))·γ_t(γ_x(τ))dτ.

	Now we define the function f:Σ→ℝ, f(x):=v(x)·γ_t(x)∈ L^1(Σ) (recall that γ_t∈ L^2𝒱(Σ)). We note that γ_x(τ)=ψ_τ(x) where ψ_τ denotes the area-preserving flow of v. Hence, standard ergodic theoretical results, c.f. <cit.>, imply that q̂∈ L^1(Σ) and that Q(v)=∫_Σf(x)dσ(x)/|Σ|=∫_Σv(x)·γ_t(x)dσ(x)/|Σ| as was to be shown.

Before we come to the proof of <Ref> we will need one additional lemma. To this end we recall that ℋ_N(Ω) denotes the space of curl-free, div-free, vector fields tangent to the boundary of a given domain Ω.

	
	Let Σ⊂ℝ^3 be a closed, connected C^1,1-surface and let Ω⊂ℝ^3 be the bounded domain enclosed by Σ, ∂Ω=Σ. Given any Γ∈ℋ_N(Ω) we let γ∈ℋ(Σ) denote the L^2(Σ)-orthogonal projection of Γ|_Σ onto the space of harmonic fields ℋ(Σ). Then ℋ(γ×𝒩)=0, where 𝒩 denotes the outward pointing unit normal.


	We follow the proof of <Ref>. Let us set γ̃:=γ×𝒩. Then
	
    4πℋ(γ̃)=∫_Σ∫_Σγ̃(x)·(γ̃(y)×x-y/|x-y|^3)dσ(x)dσ(y)
    		
    
    		=∫_Σ∫_Σ(γ̃(x)×γ̃(y))·x-y/|x-y|^3dσ(x)dσ(y)=∫_Σ∫_Σ((γ(x)×𝒩(x))×γ̃(y))·x-y/|x-y|^3dσ(y)dσ(x).

	From here on all the arguments in the computations of 4πℋ_c(γ,γ̃) apply verbatim, with the only caveat that γ(y) has to be replaced by γ̃(y) in the appropriate places during the computations. This leads us to the identity 2ℋ(γ̃)=∫_Σγ̃(y)·γ(y)dσ(y)=0 because γ̃ and γ are pointwise everywhere orthogonal to each other.

For potential future reference we also state the following immediate consequence.

	
	Let Σ⊂ℝ^3 be a closed, connected C^1,1-surface and let Ω⊂ℝ^3 be the bounded domain enclosed by Σ, ∂Ω=Σ. Given any Γ∈ℋ_N(Ω) we have Γ×𝒩∈ L^2𝒱_0(Σ) and ℋ(Γ×𝒩)=0.


	As mentioned in the proof of <Ref> we can decompose Γ|_Σ as Γ|_Σ=∇_Σf+γ for a suitable function f∈⋂_0<α<1C^1,α(Σ) and γ∈ℋ(Σ). Since taking the cross product with 𝒩 corresponds, in the language of differential forms, to applying the Hodge star operator, it is immediate that Γ×𝒩∈ L^2𝒱_0(Σ). In addition, Γ×𝒩 and γ×𝒩 differ only by a co-exact vector field, so that the corollary follows from <Ref> and <Ref>.


	Since by assumption T^2≅Σ we know from standard Hodge-theory that (ℋ(Σ))=2. Further, if we fix any Γ∈ℋ_N(Ω)∖{0} where Ω⊂ℝ^3 is the solid torus enclosed by Σ, we see that its L^2-orthogonal projection γ onto ℋ(Σ) on Σ defines a non-zero element of ℋ(Σ). Setting γ̃:=γ×𝒩 we note that γ̃∈ℋ(Σ) because taking the cross product with the outer unit-normal corresponds to applying the Hodge star operator to the corresponding 1-forms (obtained from the musical isomorphism). Now, if v∈ C^0,1𝒱_0(Σ) is an arbitrary vector field, we can decompose v by means of the Hodge decomposition theorem as v=∇_Σf×𝒩+αγ+βγ̃ for a suitable C^1-function f∈ C^1(Σ) and suitable constants α,β∈ℝ. We observe that γ and γ̃ have the same L^2-norms, so that γ̃ is L^2-normalised if γ is. Assuming that γ is L^2-normalised and keeping in mind that the Hodge decomposition is L^2-orthogonal we find
	
    α=∫_Σγ· vdσ(x), β=∫_Σγ̃(x)· v(x)dσ(x).

	Utilising <Ref> we conclude
	
    ℋ(v)=ℋ(αγ+βγ̃)=α^2ℋ(γ)+2αβℋ_c(γ,γ̃)+β^2ℋ(γ̃).

	According to <Ref> we have ℋ(γ̃)=0. In addition, we have shown in the proof of <Ref> that 2ℋ_c(γ,γ̃)=γ^2_L^2(Σ)=1 due to the normalisation of γ. We arrive at
	
    ℋ(v)=α^2ℋ(γ)+αβ.

	Now we recall that we are given a purely poloidal curve σ_p and a toroidal curve σ_t which uniquely determine elements γ_p,γ_t∈ℋ(Σ) according to the relations ∫_σ_pγ_t=0=∫_σ_tγ_p and ∫_σ_pγ_p=1=∫_σ_tγ_t. We claim that γ_t=γ/∫_σ_tγ. To see this we use the fact that Γ|_Σ-γ is the gradient of a C^1-function so that ∫_σ_pγ=∫_σ_pΓ. Further, we may assume that the C^1-curve σ_p, being purely poloidal, bounds a C^1 disc D⊂Ω, ∂ D=σ_p. The idea now is to apply Stokes' theorem to deduce ∫_σ_pΓ=∫_Dcurl(Γ)· n dσ=0 where n denotes the corresponding normal to D and where we used that Γ is curl-free. However, due to the boundary regularity, we only know that Γ∈⋂_1≤ p<∞W^1,p𝒱(Ω) so that the classical Stokes' theorem is not immediately applicable. To bypass this problem one can exploit the fact that Γ is in fact analytic within Ω, since it is a weak solution of ΔΓ=0 in Ω. We can then fix a (compactly supported) C^∞-vector field X defined on ℝ^3 which is everywhere inward pointing along Σ. If we let Ξ_τ denote the flow of X we obtain a C^1-curve σ_p(τ):=Ξ_τ∘σ_p⊂Ω which still bounds a C^1-disc within Ω since X is inward pointing. It then follows from an application of Stokes' theorem and interior regularity of Γ that ∫_σ_p(τ)Γ=0 for all τ>0. On the other hand, Sobolev embeddings tell us that Γ is continuous up to the boundary, so that it is not hard to see that ∫_σ_pΓ=lim_τ↘ 0∫_σ_p(τ)Γ=0 as previously suggested so that ∫_σ_pγ=0. Finally, if ∫_σ_tγ=0 would be true, it would follow, because σ_p and σ_t generate the first fundamental group, that ∫_σγ=0 for every closed loop σ so that a standard construction would imply that γ is a gradient field implying that γ=0 since γ is div-free and hence L^2-orthogonal to the gradient fields. Since γ≠ 0 we conclude that γ/∫_σ_tγ satisfies the defining equations of γ_t and so by uniqueness γ_t=γ/∫_σ_tγ. Using this, we obtain from (<ref>) and <Ref>
	
    α=Q(v)|Σ|∫_σ_tγ.

	Now, since γ and γ̃ form a basis of ℋ(Σ) we can also express γ_p=μγ+λγ̃ for suitable μ,λ∈ℝ. We find 1=∫_σ_pγ_p=λ∫_σ_pγ̃ so that λ=1/∫_σ_pγ̃ where ∫_σ_pγ̃≠ 0 because otherwise γ and γ̃ would be linearly dependent. Lastly, 0=∫_σ_tγ_p=μ∫_σ_tγ+∫_σ_tγ̃/∫_σ_pγ̃. We can solve this for μ and conclude from (<ref>) and <Ref>
	
    β=∫_Σγ_p/λ· v-μ/λγ· vdσ(x)=|Σ|(∫_σ_pγ̃)(P(v)+∫_σ_tγ̃/∫_σ_pγ̃Q(v))

	where we used that γ_t=γ/∫_σ_tγ. Inserting the expressions (<ref>) and (<ref>) into (<ref>) yields the result.
	
	Finally, if σ_t bounds a surface 𝒜 outside of Ω, the result follows from <Ref>.



 §.§ Proof of <Ref>


	We will divide the proof in several steps. We at first consider the divergence-free vector field w∈ C^0,1𝒱_0(Σ). We recall that by assumption there exists a positive function f∈ C^0,1(Σ,(0,∞)) such that v=w/f is rectifiable. Then in the last step we show that the rotational transform of v coincides with that of w which will conclude the proof.
	Step 1: We observe that v being rectifiable implies that either all of its field lines are closed or none of them are. Assume in the first step that the field lines of v are all closed. Then by the rectifiability property it follows further that all field lines are of the same type, i.e. if σ_x is a field line of v starting at x∈Σ and we write σ_x=P_xσ_p⊕ Q_xσ_t for suitable P_x,Q_x∈ℤ (and where σ_p,σ_t denote the fixed purely poloidal and toroidal curve respectively), then in fact P_x and Q_x are independent of the chosen point x. Since w and v have the same field lines, which are merely traced out with a different speed, we conclude that also all field lines of w are of the type Pσ_p⊕ Qσ_t with the same P,Q∈ℤ (which are independent of x and we thus drop the subscript x). We recall (<ref>) and see that if for a given x∈Σ we let τ_x>0 denote the period of the field line of w starting at x, then q̂(x)=Q/τ_x and with the same reasoning p̂(x)=P/τ_x where q̂,p̂ are the quantities associated with w. Further, by assumption, Q(w)≠ 0 which by definition implies 0≠∫_Σq̂(x)dσ(x)=Q∫_Σ1/τ_xdσ(x) and consequently Q≠ 0 so that w has a well-defined rotational transform at every x∈Σ and by definition ι_w(x)=p̂(x)/q̂(x)=P/Q since q̂(x)=Q/τ_x, p̂(x)=P/τ_x. Finally, since p̂(x)=P/Qq̂ and P/Q is a constant, we conclude P(w)=P/QQ(w) so that ι_w(x)=P(w)/Q(w) is independent of x and the statement follows from <Ref>.
	Step 2: Now we assume that the field lines of w are not closed. We recall that if we let μ denote the normalised surface measure on Σ, then an area preserving flow ψ_t is ergodic if for any (Borel-)measurable A⊂Σ with ψ_-t(A)⊂ A for all t≥ 0 we have μ(A)=0 or μ(A)=1.
	
	We claim first that the flow ψ_t of w is ergodic. Since w is div-free the flow is clearly area preserving. On the other hand, since v=w/f where f is a strictly positive function, it is easy to see that if Ψ_t denotes the flow of v, the condition ψ_-t(A)⊂ A for all t≥ 0 implies Ψ_-t(A)⊂ A for all t≥ 0. We have to prove that this implies that either A or Σ∖ A is a null-set. However, since diffeomorphisms preserve null-sets it is enough, due to the rectifiability of v, to see that the linear flow with irrational slope is ergodic on the flat torus which is a well-known fact, see <cit.>. Then the ergodicity and the ψ_t invariance of q̂ and p̂ respectively imply that q̂ and p̂ are both constant a.e.. Then by definition we find P(w)=p̂(x), Q(w)=q̂(x) for a.e. x and since by assumption Q(w)≠ 0 we see that w has a well-defined rotational transform given by ι_w(x)=p̂(x)/q̂(x)=P(w)/Q(w) so that the statement now follows once more from <Ref>.
	Step 3: Here we prove that ι_v(x)=ι_w(x) for a.e. x∈Σ, which will complete the proof.
	
	We recall the definition p̂_v(x)=lim_T→∞1/T∫_σ_x^v[0,T]γ_p, where the subscript indicates that p̂ refers to the poloidal twists of v and where σ_x^v denotes the field line of v starting at x. Further recall that w=fv for a strictly positive function f∈ C^0,1(Σ,(0,∞)). We can then consider 
	
    f̂(x):=lim_S→∞1/S∫_0^Sf(σ_x^w(t))dt,

where σ_x^w denotes the field line of w starting at x. By standard ergodic theory, c.f. <cit.>, this defines an element of L^1(Σ) since the flow of w is area-preserving. We observe that f is bounded above and away from zero so that 0≤ c_1≤f̂(x)≤ c_2<∞ for suitable constants 0<c_1≤ c_2 and all x∈Σ. Further, since v and w have the same field lines we may parametrise σ^v_x(t) via w so that we can write σ_x^v[0,T]=σ_x^w[0,S] for a suitable S>0 (in case that σ_x^v is periodic we pick S such that σ_x^w[0,S] passed x the same amount of times as σ_x[0,T] did). We therefore obtain
	
    p̂_v(x)=lim_T→∞1/T∫_σ_x^v[0,T]γ_p=lim_T→∞1/T∫_σ_x^w[0,S]γ_p

	where we note that S depends on T. To establish a relation between T and S we observe that T=∫_σ_x^v[0,T]v/|v|^2 and therefore we can write by choice of S
	
    T=∫_σ_x^w[0,S]v/|v|^2=∫_0^Sf(σ_x^w(t))dt

	where we used that w=fv. Since f is bounded away from zero and bounded above we see that T→∞⇔ S→∞. Inserting this relation into (<ref>) we find
	
    p̂_v(x)=lim_T→∞S/T·1/S∫_σ_x^w[0,S]γ_p=p̂_w(x)/f̂(x).

	In the same way one proves that q̂_v(x)=q̂_w(x)/f̂(x) so that in particular v has a well-defined rotational transform if and only if w has a well-defined rotational transform and we obtain ι_v(x)=p̂_v(x)/q̂_v(x)=p̂_w(x)/q̂_w(x)=ι_w(x) as claimed.



 §.§ Proof of <Ref>


	Case (i): If Σ≅ S^2 it follows from <Ref> that ℋ(w)=0 for all w∈ L^2𝒱_0(Σ) which implies the claim.
	Case (ii): Given any w∈ L^2𝒱_0(Σ) we can express w by means of the Hodge decomposition theorem as w=grad_Σ(f)×𝒩+γ for a suitable f∈ H^1(Σ) and γ∈ℋ(Σ). It then follows from <Ref> that if ℋ(w)≠ 0, we must have γ≠ 0. In addition, if ℋ(w)>0, the L^2(Σ)-orthogonality of the Hodge-decomposition implies that if grad_Σ(f)≠ 0 we must have ℋ(w)/w^2_L^2(Σ)<ℋ(w)/γ^2_L^2(Σ)=ℋ(γ)/γ^2_L^2(Σ) and that therefore w could not have been a maximiser. We will now at first show that there exists some w∈ L^2𝒱_0(Σ) with ℋ(w)>0 which by the previous argument will allow us to look for a maximiser within the space ℋ(Σ). To see this we let Ω denote the bounded domain bounded by Σ and fix any element Γ∈ℋ_N(Ω)∖{0} (the space of curl- and div-free H^1(Ω) vector fields which are tangent to Σ). Further, we let γ denote the L^2(Σ)-orthogonal projection of Γ|_Σ onto ℋ(Σ) and we define γ̃:=γ×𝒩. We recall that we have shown in the proof of <Ref> that 2ℋ_c(γ,γ̃)=γ^2_L^2(Σ) (ℋ_c being the cross-helicity) and that <Ref> states that ℋ(γ̃)=0. We can now let β≠ 0 be any constant and set α:=1/|β|. We define γ̂:=αγ+βγ̃ and observe that the previous considerations yield
	
    ℋ(γ̂)=α^2ℋ(γ)+2αβℋ_c(γ,γ̃)+β^2ℋ(γ̃)=α^2ℋ(γ)+αβγ^2_L^2(Σ)=1/β^2ℋ(γ)+sign(β)γ^2_L^2(Σ)

	where sign(β) equals +1 if β>0 and equals -1 if β<0. Letting β≫1 or β≪ -1 we see that there is some γ̂_± with ℋ(γ̂_±)· (±1)>0. Then we are left with observing that the functional {γ̂∈ℋ(Σ)|γ̂_L^2(Σ)=1}→ℝ,γ̂↦ℋ(γ̂) admits a (necessarily positive) global maximum since the unit sphere in finite dimensional vector spaces is compact. The claim follows then from the scaling properties of the helicity, i.e. ℋ(λ v)=λ^2ℋ(v) for all λ∈ℝ and v∈ L^2𝒱_0(Σ).



 §.§ Proof of <Ref>


	We recall that we let π:L^2𝒱(Σ)→ L^2𝒱_0(Σ) denote the L^2(Σ)-orthogonal projection from the space of square integrable vector fields on Σ into the space of square-integrable div-free fields on Σ. We prove first that π∘BS_Σ:L^2𝒱_0(Σ)→ L^2𝒱_0(Σ) is self-adjoint with respect to L^2(Σ). This is easy to see because for every v,w∈ L^2𝒱_0(Σ) we have ⟨ v,(π∘BS_Σ)(w)⟩_L^2(Σ)=⟨ v,BS_Σ(w)⟩_L^2(Σ) and one can then write out the definitions, use the cyclic property of the Euclidean inner product and Fubini's theorem to conclude the self-adjointness.
	
	We will now show that the image of π∘BS_Σ is finite dimensional and therefore this operator must be compact. Indeed, it follows from <Ref> that for any f∈ H^1(Σ) and w∈ L^2𝒱_0(Σ), we have ⟨grad_Σ(f)×𝒩,(π∘BS_Σ(w))⟩_L^2(Σ)=⟨grad_Σ(f)×𝒩,BS_Σ(w)⟩_L^2(Σ)=ℋ_c(grad_Σ(f)×𝒩,w)=0 and so the Hodge decomposition theorem implies that (π∘BS_Σ)(w)∈ℋ(Σ) for every w∈ L^2𝒱_0(Σ) which is a finite dimensional space. It now follows from the spectral theorem for compact, self-adjoint operators that π∘BS_Σ admits a discrete spectrum which accumulates at most at zero and a corresponding eigenbasis which together with Ker(π∘BS_Σ) spans the space L^2𝒱_0(Σ). According to the proof of <Ref> we see that there exist elements v,w∈ L^2𝒱_0(Σ) with ℋ(v)>0 and ℋ(w)<0. Since ℋ(v)=⟨ v,(π∘BS_Σ)(v)⟩_L^2(Σ) and similarly for w, it must be the case that π∘BS_Σ admits a positive and negative eigenvalue and since the spectrum accumulates at most at zero there must exist a largest positive and smallest negative eigenvalue. It then follows from the eigenspace decomposition that v∈ L^2𝒱_0(Σ) with v_L^2(Σ)=1 maximises helicity among all other such fields if and only if v is an eigenfield of π∘BS_Σ corresponding to the largest positive eigenvalue. Due to the scaling properties of helicity the same remains true for the maximisation of the Rayleigh quotient in (<ref>).
	
	

 §.§ Proof of <Ref>

	Before we come to the proof of <Ref> we prove the claim in <Ref>.
	
		
		Let T^2≅Σ⊂ℝ^3 be a closed, connected C^1,1-surface. Then
		
    inf_T^2≅Σ∈ C^1,1Λ(Σ)=inf_T^2≅Σ∈ C^1,1,|Σ|=1Λ(Σ) and sup_T^2≅Σ∈ C^1,1Λ(Σ)=sup_T^2≅Σ∈ C^1,1,|Σ|=1Λ(Σ).

	
	
	We first prove that for given Σ and λ>0 we have Λ(Σ)=Λ(Σ_λ) where Σ_λ:={λ x| x∈Σ}. To see this we can start with an arbitrary vector field w∈ L^2𝒱_0(Σ) and define w_λ(x):=w(x/λ) for x∈Σ_λ. One easily verifies that w_λ∈ L^2𝒱_0(Σ_λ) and that accordingly the map w↦ w_λ gives rise to an isomorphism between L^2𝒱_0(Σ) and L^2𝒱_0(Σ_λ) (note that Σ=(Σ_λ)_1/λ). Then, using the change of variable formula and noting that ψ_λ:Σ→Σ_λ, x↦λ x provides an (orientation-preserving) diffeomorphism between Σ and Σ_λ, we find w_λ^2_L^2(Σ_λ)=λ^2w^2_L^2(Σ) for all w∈ L^2𝒱_0(Σ). As for helicity, we can write, keeping in mind that we perform a double integration,
	
    ℋ(w_λ)=1/4π∫_Σ_λ∫_Σ_λw_λ(x)·(w_λ(y)×x-y/|x-y|^3)dσ(y)dσ(x)
    		
    
    		=λ^4/4π∫_Σ∫_Σw(q)·(w(p)×λ q-λ p/|λ q-λ p|^3)dσ(p)dσ(q)=λ^2ℋ(w).

	Since w↦ w_λ defines an isomorphism we obtain
	
    Λ_±(Σ_λ)=max_w̃≠ 0
    w̃∈ L^2𝒱_0(Σ_λ)±ℋ(w̃)/w̃^2_L^2(Σ_λ)=max_w≠ 0
     w∈ L^2𝒱_0(Σ)±ℋ(w_λ)/w_λ^2_L^2(Σ_λ)=max_w≠ 0
     w∈ L^2𝒱_0(Σ)±ℋ(w)/w^2_L^2(Σ)=Λ_±(Σ)

	and therefore Λ(Σ_λ)=Λ(Σ) for all λ>0. We can now take sequences (Σ_n)_n approaching either the infimum or the supremum on the left hand side of the statement of the lemma respectively and rescale the Σ_n by appropriate λ_n>0 to normalise their area which in combination with Λ(Σ_λ)=Λ(Σ) immediately implies the claim.
	
	
		We recall that according to <Ref> and <Ref> the L^2(Σ)-normalised vector fields γ_+,γ_- realising the Rayleigh quotient (<ref>) are elements of ℋ(Σ) and eigenfields of π∘BS_Σ where π:L^2𝒱(Σ)→ L^2𝒱_0(Σ) denotes the L^2(Σ)-orthogonal projection onto L^2𝒱_0(Σ). In addition, while proving the compactness of π∘BS_Σ during the course of the proof of <Ref>, we also showed that π∘BS_Σ maps L^2𝒱_0(Σ) onto ℋ(Σ). Therefore, the operator π∘BS_Σ:ℋ(Σ)→ℋ(Σ) is well-defined and the Rayleigh quotient (<ref>) is realised by some eigenfields of this restricted operator. However, the Hodge isomorphism <cit.>, implies (ℋ(Σ))=(H^1_dR(Σ))=2, where H^1_dR(Σ) denotes the first de Rham cohomology group and where we used that Σ≅ T^2. It follows further from <Ref> that π∘BS_Σ admits at least one positive and one negative eigenvalue. Consequently, π∘BS_Σ|_ℋ(Σ) is a self-adjoint operator on the 2-dimensional Hilbert space (ℋ(Σ),⟨·,·⟩_L^2(Σ)). It therefore admits precisely two eigenvalues and thus one eigenvalue is strictly positive λ_+(Σ)>0 and one eigenvalue is negative λ_-(Σ)<0 with corresponding (L^2(Σ)-orthonormal) eigenfields γ_+(Σ) and γ_-(Σ) respectively. It follows immediately from the eigenspace decomposition that Λ_+(Σ)=ℋ(γ_+)=λ_+(Σ) and Λ_-(Σ)=-ℋ(γ_-)=-λ_-(Σ)=|λ_-(Σ)|. In order to obtain a relation between λ_+(Σ) and λ_-(Σ) we will now find an explicit matrix representation of (π∘BS_Σ)|_ℋ(Σ). To this end we let Ω⊂ℝ^3 denote the bounded domain which is bounded by Σ, ∂Ω=Σ. Then we can fix any Γ∈ℋ_N(Ω)∖{0} and as usual let γ∈ℋ(Σ) denote the L^2(Σ)-orthogonal projection of Γ|_Σ onto ℋ(Σ). Upon rescaling we may assume that γ is L^2(Σ)-normalised. We then define γ̃:=γ×𝒩 where 𝒩 is the outward unit normal and note that γ̃∈ℋ(Σ) is also L^2(Σ)-normalised. We then observe that
		
    (π∘BS_Σ)(γ)=⟨ (π∘BS_Σ)(γ),γ⟩_L^2(Σ)γ+⟨ (π∘BS_Σ)(γ),γ̃⟩_L^2(Σ)γ̃=ℋ(γ)γ+ℋ_c(γ,γ̃)γ̃.

		Similarly we find (π∘BS_Σ)(γ̃)=ℋ_c(γ̃,γ)γ+ℋ(γ̃)γ̃ and therefore π∘BS_Σ has the following matrix representation with respect to the basis ℬ:={γ,γ̃}
		
    ℳ:=(π∘BS_Σ)_ℬ^ℬ=[      ℋ(γ) ℋ_c(γ,γ̃); ℋ_c(γ̃,γ)     ℋ(γ̃) ].

		It now follows from <Ref> that ℋ(γ̃)=0 and it follows from the first step in the proof of <Ref> and the symmetry of the Biot-Savart operator that 2ℋ_c(γ̃,γ)=2ℋ_c(γ,γ̃)=γ^2_L^2(Σ)=1 due to the normalisation of γ. We obtain ℳ=[ ℋ(γ)  1/2;  1/2    0 ] and conclude 
		
    λ_+(Σ)λ_-(Σ)=(ℳ)=-1/4⇔Λ_+(Σ)Λ_-(Σ)=1/4.

		Now if both Λ_+(Σ) and Λ_-(Σ) would be strictly smaller than 1/2 it would contradict (<ref>) and therefore we must have Λ(Σ)≥1/2.
		
		Lastly we observe that Λ_+(Σ)-Λ_-(Σ)=λ_+(Σ)+λ_-(Σ)=Trace(ℳ)=ℋ(γ). So if ℋ(γ)=0 we see that Λ_+(Σ)=Λ_-(Σ) and then (<ref>) becomes Λ^2(Σ)=1/4⇒Λ(Σ)=1/2. Conversely, if Λ(Σ)=1/2 we note that (<ref>) can be equivalently expressed as Λ(Σ)min{Λ_+(Σ),Λ_-(Σ)}=1/4 and that therefore min{Λ_+(Σ),Λ_-(Σ)}=1/2=Λ(Σ) and so Λ_+(Σ)=Λ_-(Σ) so that 0=Trace(ℳ)=ℋ(γ).
	
	

 §.§ Proof of <Ref>

	
		According to <Ref> we only need to prove that ℋ(γ)=0 whenever Σ is a C^1,1-rotationally symmetric torus where we recall that γ is the L^2(Σ)-orthogonal projection of any element Γ∈ℋ_N(Ω)∖{0} onto ℋ(Σ) and where Ω denotes the bounded domain with boundary Σ. Upon applying isometries to Σ we may assume that the axis of symmetry of Σ is the z-axis. One can then consider the vector field Y(x,y,z):=(-y,x,0)^tr defined on all of ℝ^3 and which induces isometries. Since Σ is a rotationally symmetric regular torus, it does not intersect the axis of symmetry so that in particular Ω does not intersect it either. One can therefore consider the vector field Γ:=Y/|Y|^2 on Ω which can be easily verified to satisfy curl(Γ)=0 and div(Γ)=0. In addition, since Y generates rotations around the z-axis, we find Y∥Σ and hence Γ∥Σ so that Γ∈ℋ_N(Ω)∖{0}. Since Γ is curl-free and tangent to Σ it is standard that Γ|_Σ is also curl-free as a vector field on Σ. Finally we note that Y induces isometries so that it is a Killing field. But the restriction of Killing fields to invariant surfaces remain Killing so that one finds that Y|_Σ is div-free. In addition, a direct calculation yields Y·grad(|Y|^2)=0 on all of ℝ^3 so that the product rule gives us div_Σ(Γ|_Σ)=0. Consequently Γ|_Σ∈ℋ(Σ) and so γ=Γ|_Σ. Lastly we note that Γ|_Σ and Y|_Σ have the same field lines which are all mutually unlinked and periodic with a uniformly bounded period. It then follows from the linking interpretation of helicity, <Ref> and <Ref>, that ℋ(γ)=0 so that the claim follows from <Ref>.
	
	

 §.§ Proof of <Ref>

	
		We recall that we call an element j∈ L^2𝒱_0(Σ) simple, if it satisfies ∫_Σj·γ_tdσ=0 or equivalently, c.f. <Ref>, Q(j)=0 where γ_t is the unique element of ℋ(Σ) satisfying ∫_σ_pγ_t=0 and ∫_σ_tγ_t=1 (recall also that the notion of being simple is independent of the chosen curves σ_p,σ_t). We have to show that for any given j∈ L^2𝒱_0(Σ) there exists some j^'∈ L^2𝒱_0(Σ) which is simple and satisfies BS(j)(x)=BS(j^')(x) for all x∈Ω, where BS(j)(x):=1/4π∫_Σj(y)×x-y/|x-y|^3dσ(y) corresponds physically to the magnetic field inside the bounded domain Ω bounded by Σ which is induced by the current density j∈ L^2𝒱_0(Σ). So let j∈ L^2𝒱_0(Σ) be given, it then follows from <cit.> that there exists some j_0∈Ker(BS)∖{0}⊂ L^2𝒱_0(Σ)∩⋂_0<α<1C^0,α𝒱(Σ) so that BS(j+α j_0)=BS(j) for every α∈ℝ. Further, it was shown in the first step of the proof of <cit.> that if we fix any Γ∈ℋ_N(Ω)∖{0}, then j_0 can be chosen such that ∫_Σj_0(y)·Γ(y)dσ(y)=Γ^2_L^2(Ω)≠ 0. Now we observe first that curl(Γ)=0 in Ω implies that Γ|_Σ is curl-free on Σ and hence we can use the Hodge decomposition theorem to express Γ=grad_Σ(f)+γ for suitable f∈⋂_0<α<1C^1,α(Σ) and γ∈ℋ(Σ). Since j_0 is div-free it is L^2(Σ)-orthogonal to all gradient fields, which yields ∫_Σj_0·γ dσ=∫_Σj_0·Γ dσ≠ 0. Finally we recall that we have shown in the proof of <Ref> that γ_t=γ/∫_σ_tγ and consequently ∫_Σj_0·γ_tdσ≠ 0. Hence, by linearity, we may always find a suitable α_j∈ℝ such that ∫_Σ(j+α_j j_0)·γ_tdσ=0 so that we may set j^':=j+α_jj_0 and the regularity claims follow immediately from the regularity of j_0. Finally, we need to show that such a j^' is unique. To this end, suppose j^',ĵ∈ L^2𝒱_0(Σ) are simple and satisfy BS(j^')=BS(ĵ). Then j̅:=j^'-ĵ is also simple and contained in the kernel of BS. Therefore j̅∈Ker(BS) and consequently there exists some β∈ℝ with j̅=β j_0 where j_0 is chosen like above. In particular, ∫_Σj_0·γ_tdσ≠ 0 so that the simplicity of j̅ implies β=0 and consequently j^'=ĵ.
	
	

 §.§ Proof of <Ref>

	
		It follows from <cit.> that for every B_T∈ L^2ℋ(P) and every ϵ>0 there exists some j∈ L^2𝒱_0(Σ) with BS(j)-B_T_L^2(P)≤ϵ. Then <Ref> implies that there exists j^'∈ L^2𝒱_0(Σ) which is simple and with BS(j)=BS(j^') which concludes the proof.
	
	

 §.§ Proof of <Ref>

	
		The proof is in essence identical to the proof of <cit.>. For convenience of the reader we present it here nonetheless. Given B_T∈ L^2ℋ(P) we can fix any ϵ>0. According to <Ref> we can find some j_ϵ∈ L^2𝒱_0(Σ) which is simple and such that BS(j_ϵ)-B_T_L^2(P)^2≤ϵ. Then, by definition, j_λ^0 is the unique global minimiser of C_0(λ;B_T) (recall (<ref>)) so that
		
    C_0(λ;B_T)≤BS(j_ϵ)-B_T^2_L^2(P)+λj_ϵ^2_L^2(Σ)≤ϵ+λj_ϵ^2_L^2(Σ)

		by choice of j_ϵ. Further, we note that
		
    BS(j_λ^0)-B_T^2_L^2(P)≤BS(j_λ^0)-B_T^2_L^2(P)+λj^0_λ^2_L^2(Σ)=C_0(λ;B_T)

		and hence
		
    BS(j^0_λ)-B_T^2_L^2(P)≤ϵ+λj_ϵ^2_L^2(Σ).

		We recall that j_ϵ depends only on B_T and ϵ but is independent of λ. Therefore we may take the limit and obtain
		
    0≤lim inf_λ↘ 0BS(j^0_λ)-B_T^2_L^2(P)≤lim sup_λ↘ 0BS(j^0_λ)-B_T^2_L^2(P)≤ϵ.

		Since ϵ>0 was arbitrary we conclude lim_λ↘ 0BS(j^0_λ)-B_T_L^2(P)=0.
	


§ ACKNOWLEDGEMENTS

I would like to thank Ugo Boscain and Mario Sigalotti for fruitful discussions about the interpretation of surface helicity, in particular for suggesting to try to artificially close the field lines as depicted in <Ref>. This work has been supported by the Inria AEX StellaCage.


§ CONTINUITY OF SURFACE BIOT-SAVART OPERATOR



	
	<Ref> is well-defined, i.e. the Biot-Savart operator indeed defines a q-integrable vector field for any choice of orthonormal frame and it is independent of the chosen frame.


	First let us suppose that we have shown that
	
    ∑_i=1^2(∫_Σ v(y)×x-y/|x-y|^3dσ(y)· E_i(x))E_i(x)

	is finite for a.e. x in the domain of definition of any fixed orthonormal frame E_i. Then, if E_i and Ẽ_j are any two orthonormal frames whose domains of definition overlap, then on this overlap the integrals will exist almost surely for any x in the overlap. For any such fixed x we then have
	
    ∑_i=1^2(∫_Σ v(y)×x-y/|x-y|^3dσ(y)· E_i(x))E_i(x)
    		
    
    		=∑_j=1^2∑_i=1^2(∫_Σ v(y)×x-y/|x-y|^3dσ(y)· E_i(x))(E_i(x)·Ẽ_̃j̃(x))Ẽ_j(x)
    		
    
    		=∑_j=1^2(∫_Σ v(y)×x-y/|x-y|^3dσ(y)·∑_i=1^2[(E_i(x)·Ẽ_̃j̃(x))E_i(x)])Ẽ_j(x)
    		
    
    		=∑_j=1^2(∫_Σ v(y)×x-y/|x-y|^3dσ(y)·Ẽ_j(x))Ẽ_j(x)

	and so the value of the Biot-Savart operator is independent of the chosen local orthonormal frames.
	
	In order to establish the q-integrability we define u(y):=𝒩(y)× v(y) and fix any w∈ T_xΣ. We observe that because v(y) is tangent to Σ we have v(y)=u(y)×𝒩(y). Then, using the vector triple product rule, we find
	
    ∫_Σ v(y)×x-y/|x-y|^3dσ(y)· w=∫_Σ(u(y)·x-y/|x-y|^3)𝒩(y)-(𝒩(y)·x-y/|x-y|^3)u(y)dσ(y)· w.

	It is well-known that there is some c>0 satisfying |x-y/|x-y|^3·𝒩(y)|≤c/|x-y| for all y,x∈Σ with y≠ x, c.f. <cit.>. On the other hand, we have, since w is orthogonal to 𝒩(x),
	
    |(u(y)·x-y/|x-y|^3)(𝒩(y)· w)|=|(u(y)·x-y/|x-y|^3)(𝒩(y)-𝒩(x))· w|≤ L|v(y)||w|/|x-y|

	where L is the Lipschitz character of 𝒩 and we used that |u(y)|=|v(y)|. Therefore, if |w|=1, then we see that there is some constant c_0>0 which is independent of v such that
	
    |∫_Σ v(y)×x-y/|x-y|^3dσ(y)· w|≤ c_0∫_Σ|v(y)|/|x-y|d σ(y).

	It follows easily from the dominated convergence theorem that if 1≤β <2 then the map Σ→ℝ, x↦∫_Σ1/|x-y|^βdσ(y) is continuous so that it is globally bounded on Σ. Hence, the above bound ensures that BS_Σ(v)∈ L^∞𝒱(Σ) for any v∈ L^p𝒱(Σ) with p>2 by an application of Hölder's inequality.
	
	If v∈ L^2𝒱(Σ), then in particular v∈ L^p𝒱(Σ) for any 1<p<2 and so the claim will follow from the case p<2 after observing that 2p/2-p converges to infinity as p↗ 2.
	
	Suppose now that 1<p<2. In that case one can use a partition of unity to localise the problem and hence reduce the situation to an application of the standard Hardy-Littlewood-Sobolev inequality, c.f. <cit.>.



§ PHYSICAL ROTATIONAL TRANSFORM


To simplify the discussion, we consider here only "unknotted" smooth solid tori Ω⊂ℝ^3 in the sense that we demand that there exists an orientation preserving C^∞-diffeomorphism Ψ:ℝ^3→ℝ^3 such that Ω=Φ(Ω_S) where Ω_S:={(x,y,z)^tr∈ℝ^3| (√(x^2+y^2)-2)^2+z^2≤1} is the standard rotationally symmetric solid torus bounded by a torus with minor radius r=1 and major radius R=2. We have the following parametrisation of Σ_S:=∂Ω_S, (ϕ,θ)↦μ(ϕ,θ)=((2+cos(θ))cos(ϕ),(2+cos(θ))sin(ϕ),sin(θ))^tr, 0≤ϕ,θ≤ 2π where ϕ is called the "toroidal" angle and θ is called the "poloidal angle". Further, we consider the following curves σ^S_p(τ):=(2+cos(τ),0,sin(τ))^tr, σ^S_t(τ):=(cos(τ),sin(τ),0)^tr, 0≤τ≤ 2π and call them the standard purely poloidal and standard purely toroidal loops. We then refer to σ_p:=Φ∘σ^S_p and σ_t:=Φ∘σ^S_t as a purely poloidal and a purely toroidal curve on Σ=∂Ω.

In addition, we observe that for any fixed ϕ∈ [0,2π), the curve σ_ϕ(τ):=μ(ϕ,τ), 0≤τ≤ 2π, is a closed circle (which bounds a disc within Ω_S). We call the family of images, Σ^ϕ_S:=σ_ϕ([0,2π]), (Σ^ϕ_S)_0≤ϕ<2π the standard foliation of Σ_S into poloidal sections. Accordingly we call (Σ_ϕ)_0≤ϕ<2π with Σ_ϕ:=Φ(Σ^S_ϕ) a foliation of Σ into poloidal sections.

Now one often assumes in plasma physics that the magnetic field lines of plasma equilibria B are transversal to the poloidal sections, i.e. never tangent to any such section, recall also <Ref> regarding the flow structure of plasma equilibria. Then, given a field line of B starting at some initial poloidal section Σ_ϕ_0, it will, due to the transversality assumption, continue to move along these poloidal sections Σ_ϕ until it returns to the initial poloidal section Σ_ϕ_0. During this toroidal turn, the field line will also have made a certain amount of poloidal turns (Δθ)_1. More precisely, we may write σ(τ)=Φ(σ_S(τ)) where σ_S(τ) is a curve on Σ_S parametrised by σ_S(τ)=μ(ϕ(τ),θ(τ)) for suitable smooth functions ϕ(τ) and θ(τ). The transversality condition demands that ϕ(τ) is a strictly increasing function in τ. Therefore there will be a unique 0<τ_1<∞ with ϕ(τ_1)=2π which corresponds to a full toroidal turn upon which we returned to Σ_ϕ_0. Then we set (Δθ)_1:=θ(τ_1). Similarly we can continue to follow our field line until we arrive again at Σ_ϕ_0 which uniquely determines a smallest time τ_1<τ_2<∞ with ϕ(τ_2)=4π. Accordingly we define (Δθ)_2:=θ(τ_2)-θ(τ_1) to be the amount of poloidal turns made during the second toroidal turn. Similarly we define (Δθ)_k to be the amount of poloidal turns during the k-th toroidal turn.

One way to define "physical rotational transform" of a magnetic field line σ starting at Σ_ϕ_0 is then as follows, c.f. <cit.>,

    ι_P(σ):=lim_n→∞∑_k=1^n(Δθ)_k/2π n

whenever the limit exists.

Suppose for now that σ is periodic, then it will be homotopic to Pσ_p⊕ Qσ_t for suitable P,Q∈ℤ. We observe that this means that σ performs precisely Q toroidal turns and P poloidal turns within one period. We are assuming (without loss of generality) that σ moves in the same toroidal direction as σ_t which corresponds to Q≥ 0 and the transversality assumption ensures Q≥ 1. We know that after every Q toroidal turns, we performed P poloidal turns so that ∑_i=1^kQ(Δθ)_i=2π kP. Thus, if kQ≤ n<(k+1)Q, we find ∑_i=1^n(Δθ)_i=2π kP+∑_i=kQ+1^n(Δθ)_i and we observe that -2π P≤∑_i=kQ+1^n(Δθ)_i≤ 2π P because we made less than Q additional toroidal turns and hence at most P poloidal turns. We note further that Q≤n/k≤ (1+1/k)Q so that n/k→ Q as n→∞. Hence, we overall arrive at

    ι_P(σ)=lim_n→∞∑_k=1^n(Δθ)_k/2π n=lim_n→∞(k/nP+R(n)/n)=P/Q

where R(n) is the remainder term which we have shown to be bounded in absolute value by 2π P. We recall that with our "mathematical" rotational transform ι we have, c.f. <Ref>, ι(σ)=∫_ΣB·γ_pdσ/∫_ΣB·γ_t dσ where B=fB and f>0 is a suitable positive function related to the pressure function of the plasma equilibrium and B is the underlying magnetic field. Most importantly, a field line σ of B starting at some point in Σ is of type Pσ_p⊕ Qσ_t if and only if the same is true for B. But then <Ref> and (<ref>) imply that ι(σ)=P/Q=ι_P(σ) so that the rotational transform agrees in this case, recall also <Ref>.

If the field line σ is not closed, we can close it artificially. The idea is that at time τ_n>0 when we return for the n-th time to the initial poloidal section Σ_ϕ_0 we close the curve by following the path σ_ϕ_0 from σ(τ_n) along the poloidal section until we reach the first point on Σ_ϕ_0 which was intersected by σ previously (recall that we will have visited Σ_ϕ_0 by that time precisely (n-1)-times resulting (due to the non-periodicity) in (n-1) intersection points and since Σ_ϕ_0 is a circle, we must necessarily arrive at such a point, provided n≥ 2). The important observation is that by means of this closing procedure we add at most an angle |α_n|≤ 2π to the total toroidal angle. If we denote by σ̃_n=σ[0,τ_n]⊕σ_ϕ_0 the corresponding concatenated path, we see that it will have made precisely n toroidal turns and P_n∈ℤ poloidal turns. Hence, if we let ι^n_P(σ):=∑_k=1^n(Δθ)_k/2π n then we obtain

    ι^n_P(σ)=P_n/n-α_n/2π n, |α_n|≤ 2π, ι_P(σ)=lim_n→∞ι^n_P(σ).

On the other hand, with the "mathematical" definition of rotational transform we find

    ι(σ)=lim_T→∞∫_σ[0,T]γ_p/∫_σ[0,T]γ_t

and know that this limit exists for a.e. x∈Σ at which the field line σ starts and that the limit is independent of the chosen sequence T→∞, c.f. <Ref>. We can therefore consider the sequence T=τ_n which diverges to +∞ as n→∞. But we observe that ∫_σ[0,τ_n]γ_p=∫_σ̃_nγ_p-∫_σ_ϕ_0[t^1_n,t^2_n]γ_p=P_n-r_n where we define the remainder r_n:=∫_σ_ϕ_0[t^1_n,t^2_n]γ_p and where we used that σ̃_n is closed and makes P_n poloidal turns. We observe further that |r_n| can be uniformly bounded because γ_p is globally bounded on Σ, σ̇_ϕ is globally bounded along any poloidal section and because by construction t^2_n-t^1_n=|α_n|≤ 2π is the additional toroidal angle which we need to sweep out to close our curve. Identically we can argue that ∫_σ[0,τ_n]γ_t=n+R_n where R_n is a uniformly (in n) bounded error term and we made use of the fact that σ̃_n makes n toroidal turns. We therefore arrive at

    ∫_σ[0,τ_n]γ_p/∫_σ[0,τ_n]γ_t=P_n-r_n/n+R_n=P_n/n-r_n/n/1+R_n/n=ι^n_P(σ)+α_n/2π n-r_n/2π/1+R_n/n.

Since the limit on the left hand side exists we conclude that also the physical rotational transform is well-defined and that ι(σ)=lim_n→∞∫_σ[0,τ_n]γ_p/∫_σ[0,τ_n]γ_t=lim_n→∞ι^n_P(σ)=ι_P(σ) and therefore the physical rotational transform and mathematical rotational transform always coincide and hence <Ref> provides a way to compute the rotational transform of a plasma equilibrium B, see also <Ref>.


§ EIGENFIELDS OF THE SURFACE BIOT-SAVART OPERATOR



	
	Let T^2≅Σ⊂ℝ^3 be a closed, connected C^1,1-torus and let Ω⊂ℝ^3 be the unique bounded domain whose boundary is Σ. Let further Γ∈ℋ_N(Ω)∖{0} and let γ denote the L^2(Σ)-orthogonal projection of Γ|_Σ onto ℋ(Σ) and set γ̃:=γ×𝒩 where 𝒩 denotes the outward unit normal. Further we scale Γ such that γ is L^2(Σ)-normalised. Then the set of vector fields v_±∈ℋ(Σ) realising the Rayleigh quotient Λ_±(Σ), c.f. (<ref>), are 1-dimensional subspaces E_±(Σ) of ℋ(Σ) respectively and spanned by the following vector fields
	
    E_+(Σ)=span{(√(ℋ^2(γ)+1)+ℋ(γ))γ+γ̃}, E_-(Σ)=span{γ-(√(ℋ^2(γ)+1)+ℋ(γ))γ̃}

	and we have
	
    Λ_+(Σ)=√(ℋ^2(γ)+1)+ℋ(γ)/2, Λ_-(Σ)=√(ℋ^2(γ)+1)-ℋ(γ)/2.



	We recall that we have shown in the proof of <Ref> that we may restrict attention to the operator (π∘BS_Σ)|_ℋ(Σ) which maps into ℋ(Σ) and has the following matrix representation with respect to the basis ℬ:={γ,γ̃}
	
    ℳ:=(π∘BS_Σ)^ℬ_ℬ=[ ℋ(γ)  1/2;  1/2    0 ].

	Computing the eigenvalues and eigenvectors of this matrix is a standard exercise and yields the result upon recalling that Λ_+(Σ)=λ_+(Σ) and Λ_-(Σ)=-λ_-(Σ) where λ_+(Σ),λ_-(Σ) denote the positive and negative eigenvalue of ℳ respectively and that the eigenvectors of ℳ precisely coincide with the vector fields v_± of interest.


	We note that we have already argued in the proof of <Ref> that ℋ(γ)=0 whenever Σ is a rotationally symmetric C^1,1-torus. In addition, upon applying isometries and therefore assuming that the z-axis is the symmetry axis, we had found that γ=Γ|_Σ where Γ=Y/|Y|^2 and Y(x,y,z)=(-y,x,0)^tr it the vector field inducing rotations around the z-axis. Hence <Ref> tells us that in this case Λ_±(Σ)=1/2 and that E_±(Σ)={γ±γ̃} where γ is explicitly known and γ̃=γ×𝒩 is determined by the outward unit normal field of Σ which can for instance be obtained from a globally defining function of Σ.

The downside of <Ref> is that we still need to compute ℋ(γ) which is again a non-trivial task. The following proposition reduces the problem to compute certain (simple) line and surface integrals.

	
	Let T^2≅Σ⊂ℝ^3 bound a C^1,1-solid torus Ω⊂ℝ^3 and suppose that σ_t is a toroidal curve which bounds a bounded C^1,1-surface 𝒜 outside of Ω, i.e. 𝒜⊂ℝ^3∖Ω and ∂𝒜=σ_t. Let Γ∈ℋ_N(Ω)∖{0} be fixed and let γ denote the L^2(Σ)-orthogonal projection of Γ|_Σ onto ℋ(Σ). Further let γ̃:=γ×𝒩. Then we have
	
    ℋ(γ)=-∫_σ_tγ̃/∫_σ_tγγ^2_L^2(Σ).



	We follow the ideas of the proof of <Ref>. We observe that we want to compute 4πℋ(γ)=4πℋ_c(γ,γ). We recall that γ̃=γ×𝒩 and consequently γ=𝒩×γ̃. We can then follow the reasoning of Step 1 in the proof of <Ref> where we computed ℋ_c(γ̃,γ). We observe that since γ=-(γ̃×𝒩) we will get an additional minus sign in the final expression. Apart from that only γ(x) has to be replaced by γ̃(x) throughout the computations which eventually leads us to the following identity
	
    4πℋ(γ)=-∫_Σγ(y)·[4πγ̃(y)/2+∫_Ωdiv(ṽ)(x)x-y/|x-y|^3d^3x+∫_Ωcurl(ṽ)(x)×x-y/|x-y|^3d^3x]dσ(y)
    		
    
    		=-∫_Σγ(y)·[∫_Ωdiv(ṽ)(x)x-y/|x-y|^3d^3x+∫_Ωcurl(ṽ)(x)×x-y/|x-y|^3d^3x]dσ(y)

	where we used that γ and γ̃ are perpendicular and where ṽ is any H^1-vector field on Ω such that ṽ^∥=γ̃. We note that if 𝒩×ṽ=γ, then due to the relation of γ and γ̃ this implies ṽ^∥=γ̃. It then follows from <cit.> and its proof and the regularity of γ that there exists some ṽ∈ W^1,2𝒱(Ω) satisfying 𝒩×ṽ=γ, div(ṽ)=0 in Ω and curl(ṽ)=Γ where Γ∈ℋ_N(Ω) satisfies Γ^2_L^2(Ω)=∫_Σγ·Γdσ. We note that in particular ∫_Σγ·Γdσ≥ 0 and since ∫_Σγ·Γ dσ=γ^2_L^2(Σ)>0 we must have, since ℋ_N(Ω) is spanned by Γ, Γ=Γ_L^2(Ω)/Γ_L^2(Ω)Γ from which we conclude Γ_L^2(Ω)=∫_Σγ·Γdσ/Γ_L^2(Ω)=∫_Σγ·Γ dσ/Γ_L^2(Ω). We hence arrive at curl(ṽ)=∫_Σγ·Γ dσ/Γ^2_L^2(Ω)Γ=γ^2_L^2(Σ)/Γ^2_L^2(Ω)Γ where we used that γ is the L^2(Σ)-orthogonal projection of Γ|_Σ onto ℋ(Σ).
	
	Combining these considerations, we arrive at
	
	
	ℋ(γ)=γ^2_L^2(Σ)/Γ^2_L^2(Ω)∫_Σγ(y)·BS_Ω(Γ)(y)dσ(y)

	where BS_Ω(Γ)(y)=1/4π∫_ΩΓ(x)×y-x/|y-x|^3d^3x denotes the volume Biot-Savart operator.
	
	Since Γ is div-free and tangent to the boundary of Ω it is standard, <cit.>, that curl(BS_Ω(Γ))=Γ in Ω and curl(BS_Ω(Γ))=0 outside of Ω. We further observe that since curl(BS_Ω(Γ))=Γ∥Σ, the tangent part of the restriction of BS_Ω(Γ) to Σ is curl-free. In particular, according to the Hodge decomposition theorem it follows that the L^2(Σ)-orthogonal projection of (BS_Ω(Γ))^∥ onto ℋ(Σ) differs from the original tangential field (BS_Ω(Γ))^∥ only by a gradient field. Consequently, if we let σ_t denote the toroidal curve bounding a surface 𝒜 outside of Ω, we obtain, by virtue of Stokes' theorem,
	
		
		∫_σ_tπ_ℋ(Σ)((BS_Ω(Γ))^∥)=∫_σ_tBS_Ω(Γ)=∫_𝒜curl(BS_Ω(Γ))·n dσ=0,
	
	where π_ℋ(Σ) denote the projection onto ℋ(Σ), where we used that σ_t is tangent to Σ, where n denotes the corresponding unit normal and where we used that curl(BS_Ω(Γ))=0 on ℝ^3∖Ω.
	
	Due to the scaling properties of helicity we may now without loss of generality assume that γ (and therefore γ̃) are L^2(Σ)-normalised and hence form an L^2(Σ)-orthonormal basis of ℋ(Σ). We can then express
	
		
		π_ℋ(Σ)((BS_Ω(Γ))^∥)=∫_Σγ·BS_Ω(Γ)dσγ+∫_Σγ̃·BS_Ω(γ)dσγ̃
	
	where we used that γ and γ̃ are tangent to Σ. Combining (<ref>),(<ref>) and (<ref>) (keeping in mind that we assume γ to be normalised) we arrive at
	
		
		0=∫_σ_tπ_ℋ(Σ)((BS_Ω(Γ))^∥)=Γ^2_L^2(Ω)ℋ(γ)∫_σ_tγ+∫_Σγ̃·BS_Ω(Γ)dσ∫_σ_tγ̃.
	
	Finally we recall that (BS_Ω(Γ))^∥ is curl-free in the weak sense on Σ and that so is Γ|_Σ because curl(BS_Ω(Γ))=Γ and curl(Γ)=0 in Ω are both tangent to Σ. Accordingly Γ×𝒩 and γ×𝒩=γ̃ differ only by a co-exact vector field and keeping in mind that (BS_Ω(Γ))^∥ is L^2(Σ)-orthogonal to the co-exact fields (as it is weakly curl-free on Σ) we may calculate
	
		∫_Σγ̃·BS_Ω(Γ)dσ=∫_Σγ̃·(BS_Ω(Γ))^∥dσ=∫_Σ(Γ×𝒩)·(BS_Ω(Γ))^∥dσ=∫_Σ(Γ×𝒩)·BS_Ω(Γ)dσ
		
		
		=∫_Σ(BS_Ω(Γ)×Γ)·𝒩dσ=∫_Ωdiv(BS_Ω(Γ)×Γ)d^3x=Γ^2_L^2(Ω),
	
	where we used the cyclic properties of the Euclidean product, that γ̃ it tangent to Σ and the standard calculus formula div(X× Y)=curl(X)· Y-X·curl(Y) for arbitrary C^1-vector fields X,Y on Ω. We can insert this identity in (<ref>) and finally arrive at
	
    0=Γ^2_L^2(Ω)(ℋ(γ)∫_σ_tγ+∫_σ_tγ̃)

	which immediately implies the claimed identity.

The following final result allows us to express the helicity of a vector field through Γ alone.

	
	Let T^2≅Σ⊂ℝ^3 bound a C^1,1-solid torus Ω⊂ℝ^3. Suppose that σ_p is a purely poloidal curve which bounds a C^1,1-disc D⊂Ω and suppose that σ_t is a toroidal curve bounding a C^1,1-surface 𝒜⊂Ω^c. Let further Γ∈ℋ_N(Ω)∖{0} and let γ denote the projection of Γ|_Σ onto ℋ(Σ) and set γ̃:=γ×𝒩. Then
	
		
		∫_σ_tγ=∫_σ_tΓ, ∫_σ_pγ̃=Flux(Γ)/Γ^2_L^2(Ω)γ^2_L^2(Σ)
	
	where Flux(Γ):=∫_DΓ· n dσ and n is the unit normal on the disc D compatible with a chosen orientation of σ_p with regards to Stokes' theorem. In particular, if v∈ C^0,1𝒱_0(Σ), then
	
		
		ℋ(v)/|Σ|^2=Q(v)P(v)(∫_σ_tΓ)Flux(Γ)/Γ^2_L^2(Ω)=∫_Σv·Γdσ∫_Σv·BS_Ω(Γ)dσ/Γ^2_L^2(Ω)|Σ|^2.
	


	The first identity in (<ref>) follows from the fact that Γ|_Σ is curl-free in the weak sense and therefore Γ|_Σ and γ differ only by a gradient field, so that the fact that σ_t is a closed curve yields the result. We assume now without loss of generality that γ is L^2(Σ)-normalised. Then for the second identity in (<ref>) we may utilise (<ref>),(<ref>) and the fact that ∫_σ_pγ=0 to obtain
	
    Flux(Γ)=∫_DΓ· ndσ=∫_Dcurl(BS_Ω(Γ))· ndσ=∫_σ_pBS_Ω(Γ)=∫_σ_pπ_ℋ(Σ)((BS_Ω(Γ))^∥)=Γ^2_L^2(Ω)∫_σ_pγ̃

	where we used that D⊂Ω, ∂ D=σ_p, that σ_p is tangent to Σ and that (BS_Ω(Γ))^∥ is curl-free in the weak sense.
	
	The first equality in (<ref>) is a direct consequence of (<ref>) and <Ref>. As for the second identity we recall that we have argued on several occasions that (∫_σ_tγ) γ_t=γ and we have seen that ∫_σ_pπ_ℋ(Σ)((BS_Ω(Γ))^∥)=Flux(Γ) and according to (<ref>) ∫_σ_tπ_ℋ(Σ)((BS_Ω(Γ))^∥)=0 so that by the defining properties of γ_p we find Flux(Γ)γ_p=π_ℋ(Σ)((BS_Ω(Γ))^∥). We can then utilise <Ref> and use the fact that v is tangent to Σ and div-free and that Γ and γ as well as (BS_Ω(Γ))^∥ and π_ℋ(Σ)((BS_Ω(Γ))^∥) only differ by a gradient field respectively which yields the last equality.

plain



 Paper Title: Affine Connection Approach to the Realization of Nonholonomic Constraints by Strong Friction Forces.



Affine Connection Approach to the Realization of Nonholonomic Constraints by Strong Friction Forces]Affine Connection Approach to the Realization of Nonholonomic Constraints by Strong Friction Forces










vaughngzenda@cmail.carleton.ca

Robin Chhabra*robin.chhabra@carleton.ca

Department of Mechanical and Aerospace Engineering, Carleton University, 1125 Colonel By Dr, Ottawa, K1S 5B6, Ontario, Canada






   In this paper, we study an affine connection approach to realizing nonholonomic mechanical systems mediated by viscous friction forces with large coefficients, viewed as a singular perturbation of the nonholonomic system. We show that the associated slow manifold is represented coordinate-free as the image of a section over the nonholonomic distribution. We propose a novel invariance condition based on covariant derivatives and prove that this condition is equivalent to the classical invariance condition based on time derivatives. Accordingly, we propose a novel recursive procedure to approximate the slow manifold based on the covariant derivatives of a formal power series expansion of the section. Using this recurrence relation, we derive, up to second order, approximations of the slip velocities residing in the slow manifold, as well as the associated approximated dynamics up to first order. Lastly, we illustrate our approach with a case study of a vertical rolling disk. 








[
    *
    August 5, 2024
==================

Submitted to the Journal of Nonlinear Dynamics




§ INTRODUCTION

Rolling without slipping is a constraint that appears in many mechanical systems, such as vehicle systems, and is commonly viewed as an example of an ideal nonholonomic constraint <cit.>. Nonholonomic constraints normally refer to point-wise linear constraints on the velocity space at each configuration, which cannot be described by a derivative of a set of constraints on the configuration space <cit.>. Hence, such constraints are then viewed as a nonintegrable linear distribution of the tangent bundle to the configuration manifold <cit.>. However, the nonholonomic constraint forces are only valid for velocities in the distribution. In practice, nonholonomic constraints in vehicle systems are often violated, resulting in an accumulation of errors for standard nonholonomic model-based control laws. To model the velocities violating the nonholonomic constraints, the ideal nonholonomic constraint forces may be modelled as dissipative friction, whose limit at large coefficients approximates the ideal nonholonomic system <cit.>. The large friction coefficients introduce a steep linear relationship between the friction force and the motion of the contact point that is analogous to the linear regime of Bakker-Pacejka magic formula <cit.>. Mathematically, the system with large coefficients may be treated as a slow/fast singular perturbation where the dissipative friction forces act on a much faster time scale than the nominal variables resulting in an exponentially attractive invariant manifold. This manifold – often called the slow manifold – physically represents small violations of the nonholonomic constraints. 

Nonholonomic mechanical systems form a class of constrained systems whose velocity constraints have a natural differential geometric structure  <cit.>. Classically, the equations of motion are derived using Lagrange-d'Alembert's (L-d'A) variational principle – which is an adaptation of Hamilton's variational principle –  whereby the variations are constrained to the nonholonomic distribution and enforced with Lagrange multipliers that do no work on the distribution <cit.>. Some examples of nonholonomic constraints include rolling without slipping, no sliding, and conserved angular momenta, among
others <cit.>. The kinetic energy of the mechanical system may be interpreted geometrically as a Riemannian metric tensor <cit.>. Equivalent to the Lagrange-d'Alembert principle, Lewis and Bullo <cit.> use Riemannian geometry to model the nonholonomic dynamics with affine connections. An affine connection is a geometric object that enables a notion of differentiation of vector fields on a smooth manifold. A unique affine connection is derived from the metric tensor called the Levi-Civita affine connection that encodes Coriolis, centrifugal, and centripetal accelerations. In <cit.>, the nonholonomic constraints may be viewed as the kernel of a projection map onto the complement of the nonholonomic distribution. In the affine connection framework, <cit.>, the nonholonomic constraints are then enforced using Lagrange multipliers that are solved for using a covariant derivative of the complementary projection map. As such, Lewis introduces an affine connection called the nonholonomic connection, based on the Levi-Civita connection and the covariant of the orthogonal projection map, modelling the necessary accelerations required to maintain the nonholonomic constraints. The geometric interplay of affine connections and nonintegrable distributions is studied in detail in <cit.>. Other equivalent approaches to nonholonomic mechanics include symplectic geometry <cit.>, Poisson geometry <cit.>, Gauss's Principle of Least Constraint <cit.> and Kane's equations <cit.>.

Singularly perturbed systems (SPS) manifest in various domains of science and engineering, emerging when a dynamical system relies on scalable parasitic parameters like small time constants, masses, friction, . Accordingly, SPS have natural applications in electric circuits <cit.>, chemistry <cit.>, neuroscience <cit.>, among others. Slow/fast systems are a particular case of a SPS that involve two distinct time scales, delineated by a singular parameter ϵ. Tikhonov established that when the spectra of the linearized fast dynamics lies in the left half-plane, the flow converges to an invariant (slow) manifold as ϵ tends to 0 <cit.>. The functional relationship between the slow and fast variables is dictated by an invariance condition and a formal power series expansion is a common approach to approximate the slow manifold <cit.>. Fenichel examined the persistence of the invariant manifold under small perturbations of , utilizing the concept of Normally Hyperbolic Invariant Manifold (NHIM) and under the assumption of compactness in the state space <cit.>. In <cit.>, Fenichel utilizes a geometric approach to investigate asymptotic expansions of the graph representation of the invariant manifold. In the Riemannian geometric context, Eldering extends Fenichel's findings to non-compact state spaces, assuming regularity under bounded geometry conditions <cit.>. Eldering's work opens up a pathway for the coordinate-independent examination of such systems. If a mechanical system possesses a noncompact configuration space Q and is endowed with a symmetry group G, where the group action leads to a compact shape space Q/G <cit.>, then it follows that Q has bounded geometry <cit.>. In practical applications, numerous mechanical systems exhibit symmetry groups, as seen in mobile vehicles that yield compact shape spaces <cit.>. Hence, Eldering's geometric singular perturbation techniques <cit.> can be employed for the analysis of such systems.


The physical realization of nonholonomic mechanical systems by strong viscous friction forces was treated classically by Brendelev and Karapetian who mainly focused on convergence to the ideal nonholonomic dynamics <cit.>. More recently, Eldering used the geometric singular perturbation theory of NHIM's <cit.> to generalize these convergence results for systems with bounded geometry. Common to all these models, the contact forces are modelled as a Rayleigh dissipation function (RDF) defined by a symmetric (0,2)-tensor. Rayleigh dissipation functions
have been used to model surface contact forces following first physical principles <cit.>. To realize the nonholonomic constraints, the RDF is constructed such that its kernel is identified with the nonholonomic distribution ensuring that the contact forces do no work on the distribution, mimicking the L-d'A principle, yet the dynamics are exponentially attractive to the distribution. At the limit of large friction coefficients, Eldering proves the existence of a persistent exponentially attractive NHIM and shows that in a nonstandard frame spanning the distribution and its complement, the slow manifold may be represented as the graph over the nonholonomic variables <cit.>. It is shown that this graph may be represented as a formal power series, which is used to approximate the slow manifold up to first order <cit.> based on the classical invariance condition.

Antali and Stepan considered the physical realization of the rolling phenomenon of wheels by dry friction forces <cit.>. Their analysis is based on the crossing dynamics of extended Filippov systems <cit.>. A Filippov system is described by a piece-wise smooth vector field with discontinuities on a discontinuity manifold <cit.>. Extended Filippov (EF) systems are Filippov systems with codimension-2 discontinuity manifolds and describe Coulomb rigid body dynamics <cit.>. The EF approach has been used to analyze the spinning and sliding behavior of rigid bodies under contact <cit.>. Recently, bifurcation analysis of EF systems has also been developed to analyze slip and sliding dynamics of a wheel motion <cit.>. In contrast to the slow-fast realization by viscous friction forces <cit.>, the Coulomb friction in <cit.> is both nonsmooth and nonlinear in velocities. The wheel rolling phenomenon has also been captured as a limit of piece-wise continuous holonomic constraints <cit.>. Other historical approaches to the realization of nonholonomic constraints include coupling of an external force to generate nonholonomic dynamics <cit.>.



In this paper, we study realizations of nonholonomic systems as slow/fast systems using an affine connection approach inspired by the work of Bullo and Lewis <cit.>. In our geometric approach, we represent the slow manifold coordinate-independent as the image of a section over the nonholonomic distribution. We introduce a novel invariance based on a covariant derivative along the integral curves on the slow manifold. We prove that this novel invariance condition is equivalent to the invariance condition based on time derivatives <cit.>. We propose a recursion to approximate the velocities in the slow manifold based on covariant derivatives of geometric quantities. Using the recursion, we derive approximations of equations of motion. Lastly, we study an example system of a vertical rolling disk. The major contributions of this work are summarized as follows:

    
  * formulating a coordinate-free representation of the slow manifold as the image of a nonlinear section over the nonholonomic distribution, 
    
  * formulating a novel invariance condition based on covariant derivatives, and proving its equivalence to the classical invariance condition, 
    
  * proposing a novel recurrence relation to approximate the slip velocities described by the slow manifold,
    
  * using the recurrence relation to derive higher-order perturbations of the equations of motion.


The paper is organized as follows. Section <ref> provides a self-contained summary of the differential geometric structures used in the remainder of the article. This includes a review of Riemannian geometry, affine connection systems, nonholonomic affine connection systems, and the geometry of the second tangent bundle including the procedure to take covariant derivatives of general (nonlinear) tangent bundle maps. In Section <ref>, we derive the equations of motion for a slow fast realization of nonholonomic systems and represent the slow manifold as the image of a section. Using this model, in Section <ref>, we derive the relationship between the reaction forces and the slow manifold and calculate zeroth, first and second order approximations of the slip velocities, and zeroth and first-order dynamical equations of motion. Lastly, in Section <ref> we consider our approach with a case study of a vertical rolling disk. 








§ PRELIMINARIES
 
This section provides a self-contained review of mathematical notions used throughout the paper. For references on Riemannian geometry, geometric mechanics and related topics see <cit.> and references therein. 


 §.§ Riemannian Geometry and Affine Connection Systems


Let Q be an n-dimensional smooth manifold representing the configuration space of a simple mechanical system. Let TQ = _q∈ QT_qQ and T^*Q = _q∈ QT_q^*Q be the disjoint union of tangent and cotangent spaces called the tangent and cotangent bundle, respectively. Let π_TQ:TQ → Q be the bundle projection map for TQ and let TQ be the set be the set of vector fields on Q. In local coordinates (U, = (q^1,…,q^n)), we denote the coordinate induced basis by {q^1,…, q^n} for T_Q and { q^1,…, q^n} for T^*_Q. The natural pairing .|.:T^*_Q× T_Q → between vectors and covectors in the tangent and cotangent spaces, respectively, is defined by 

    _|_ = ∑^n_i=1_i X^i,   _∈ T^*_Q,  _∈ T_Q.
 where _ = ∑_iX^iq^i and _ = ∑_i_i q^i in the canonical basis of the tangent and cotangent bundles, having components X^1,…,X^n ∈ and _1,…,_n ∈, respectively.
 
Let _:T_Q × T_Q → be a Riemannian metric on Q defining the kinetic energy of the system <cit.>. For each ∈ Q the metric  induces two isomorphisms between the tangent and cotangent bundles given point-wise by 

    ♯_:T^*_Q→ T_Q
       _↦(_^♯_,_) = _|_,  ∀_∈ T_Q
        
          ♭_:T_Q→ T^*_Q
       _↦_q^♭_|_ = _(_,_),  ∀_∈ T_Q
 called the sharp and flat isomorphisms, respectively <cit.>. In the canonical local coordinates for TQ and T^*Q, the metric tensor takes the matrix form such that by abuse of notation we denote the matrix by _. As the result, the sharp and flat isomorphisms become _^♯_ = _^-1_ and ^♭__ = __, where _ and _ are expressed in the canonical bases. 

The Levi-Civita connection , is the unique affine connection associated with  and is derived from the assumption that the connection is divergentless and torsion-free <cit.>. The ∇_ be the covariant derivative of a vector field  along a vector field  with respect to ∇ is given by 

    = (Y^iq^jX^j + Γ^i_jkX^jY^k )q^i,
 where the Levi-Civita connection coefficients – also called the Chistoffel symbols– Γ^i_jk in the coordinate-induced basis is expressed as 

    Γ^i_jk = 1/2^il(_jlq^k + _klq^j - _jkq^l).
 The Christoffel symbols have the physical meaning for a simple mechanical system of Coriolis, centripetal and centrifugal accelerations <cit.>. The action of  on (1,1)-tensors ∈(T^*Q ⊗ TQ) (such as a projection map) is given by

    () = (^i_jq^kX^k + Γ^i_kl^l_jX^k - Γ^l_kj^i_lX^k)q^i⊗ q^i.
  We may consider the connection coefficients as the point-wise bilinear map Γ:T_Q× T_Q → T_Q as the function (_,_)↦Γ(,_)_. 

The Lagrangian L:TQ → for a simple mechanical system is defined as 

    L = 1/2_(_,_) - V().
 The Euler-Lagrange equations of motion for Lagrangian L are equivalent to forced affine connection system (∇,,V,F) 

    (t)(t) = (- V + F)^♯_,   t(t) = (t).
 defined by the connection and the gradient of the potential and external force F ∈ T^*Q <cit.>. 







  §.§.§ Nonholonomic Affine Connection System


We review the basic results of nonholonomic affine connection systems treated. For general references, see <cit.>.

Let {a^l⊂ T^*Q |  l=1,…, m} be a collection of everywhere linearly independent one-forms over Q such that they specify an (n-m)-dimensional nonholonomic distribution  at each point ∈ Q by 

    _ = {_∈ T_Q  |  a^l_|_ = 0,   l =1,…,m }.
 The constraint one-forms are locally given by a^l_ = ∑_i= 1^n^l_i() q^i for each l = 1,…,m. A basis for _ is chosen by selecting m everywhere linearly independent vector fields, {_1(),…,_n-m()} satisfying the constraint equation (<ref>). Using the Riemannian metric we complete the basis for T_Q by defining 

    ^_ := {(a^1_)^♯_,…, (a^p_)^♯_}.
 
Therefore, each tangent space can be -orthogonally decomposed into T_Q ≅_⊕^_. Define the projection maps _:T_Q →_ and ^_:T_Q →^_ at each point ∈ Q. The nonholonomic distribution  can be described in terms of the projection maps as

    = {∈ TQ   |   = (),  ∈ TQ} = {∈ TQ   |  ^() = }.

 To represent the evolution of the trajectory of the system :[a,b]→ Q, we form the nonholonomic affine connection system:

    _(t)(t) = (t) -( V )^♯_, 
       ∈^((t)) = , 
       t(t) = (t),
 where  is a section in ^ representing the Lagrange multipliers representing the constraint reaction forces given by

    (t) = - (_^)() + ^( V )^♯_.
 


    The nonholonomic affine connection <cit.> is obtained by substituting the Lagrange multipliers (<ref>) into the (<ref>) and restricting the equations of motion (<ref>)-(<ref>) to the distribution .








 §.§ Geometry of the Second-Tangent Bundle

In this section, we review the geometry of the second tangent bundle TTQ. First, we review the bundle structure and decomposition <cit.> which is required to compute the covariant derivative of a general nonlinear vector bundle map. The covariant derivative of a general nonlinear vector bundle map is a prerequisite for the main results of this article. It is noteworthy that this specific result isn't explicitly stated in the literature. Lastly, we review the second tangent bundle in the context of geometric mechanics <cit.>. For general references, see <cit.>. 



  §.§.§ Vertical/Horizontal Decomposition of TTQ

Consider the second tangent bundle of Q with bundle structure given by π_TTQ:TTQ → TQ. Local coordinates for T_(,_)TQ by (,_,_(,_),_(,_)) where (_(,_),_(,_)) given in the canonical coordinate induced basis by

    _(,_) = ∑_i=1^nV^iq^i |_(,_),   _(,_) = ∑_j=1^nW^jv^j |_(,_).
 where q^i :=π^*_TQq^i. The push-forward of the projection map π_TQ induces a bundle structure (π_TQ)_*:TTQ → TQ. In these coordinates, the bundle projection maps satisfy 

    π_TTQ: (,,,) ↦(,)
       (π_TQ)_*: (,,,) ↦(,).
 The vertical bundle is defined by TQ = (π_TQ)_* and given locally by 

    TQ    = {(,,,) ∈ TTQ  |  (π_TQ)_*(,,,) = }
       = {v^i |_(,_)  |   i = 1,…,n }.
 
The parallel transport equation induces a vector-valued connection one form ω∈Ω^1(TQ;End(^n)) given by (see section 9.7 of <cit.>)

    ω :=  v^k + Γ^k_ijv^j x^i.
 Define the horizontal distribution TQ = (ω) ⊂ TTQ  <cit.> which is given locally by 

    TQ = i =1,…,n{q^i |_(,_) - Γ^k_ijv^jv^k |_(,_)}.
 Therefore, the second tangent bundle is decomposed as TTQ ≅ TQ≅ TQ. Define the vertical lift <cit.> of a vector field ∈ TQ to the vertical space _(,_)TQ

    : TQ →_(,_)TQ
       ↦t(_ + t) |_t=0 .
 In coordinates, the vertical lift is given by ^ = X^iv^i. The vertical lift is a smooth bijection and hence provides an isomorphism between the _(,_)TQ and TQ given by := ()^-1

    :_(,_)TQ →  TQ
       (,_,,) ↦(,).
 Similarly, we define the horizontal lift <cit.> with respect to the affine connection  with Christoffel Symbols Γ by

    : T_qQ× TQ →_(,_)TQ
       (,_,) ↦(,_,,- Γ(,_)).
 and its inverse := ()^-1 given by the map

    :  _(,_)TQ → TQ
       (,_,, - Γ(,_)) ↦(,).
 Lastly, we define projection maps from the tangent of the tangent bundle onto its vertical and horizontal subbundles. The projection onto the vertical bundle is defined intrinsically by ^_Γ = id_ TQ⊕Γ and given locally by 

    ^_Γ: TTQ → TQ
       (,_,,) ↦(,_,,  + Γ(,_)).
 From the projection property, the projection onto the horizontal space TQ is defined intrinsically as _Γ^:= id_TTQ - _Γ^ and given locally by 

    _Γ^: TTQ→_(,_)TQ
       (,_,,) →(,_,,-Γ(,_)).
 From these projection maps, we define projection from TTQ to two copies of TQ, each of which is isomorphic to TQ and TQ, respectively. These projections are defined in the following diagram:


    

    TTQ       𝒱TQ       TQ 
    
    	TTQ       ℋ TQ       TQ
    	["𝒬^𝒱_Γ"', from=1-1, to=1-3]
    	["↓_ver"', from=1-3, to=1-5]
    	["_𝒱", curve=height=-18pt, from=1-1, to=1-5]
    	["𝒬^ℋ_Γ", from=2-1, to=2-3]
    	["↓_hor", from=2-3, to=2-5]
    	["_ℋ"', curve=height=18pt, from=2-1, to=2-5]


 Therefore, the isomorphism TTQ ≅π_TQ^*TQ⊕π_TQ^*TQ is established by _⊕_. 

    The projection maps _:TTQ → TQ and _:TTQ → TQ satisfy the following properties: 
    
        
  * _  = (π_TQ)_*
        
  * _(_*) =, for a section :Q → TQ.
    


    See appendix <ref>



  §.§.§ Covariant Derivative of a Tangent Bundle Map

In this subsection, we find the covariant derivative of a general (nonlinear) tangent bundle map. This covariant derivative is essential for the results in subsequent sections. The authors have not found this result explicitly stated in the literature. Lastly, we provide a matrix formulation of the equations for practical computations.  

 
Let :TQ → TQ be a general (not necessarily linear) vector bundle map over the tangent bundle π_TQ:TQ → Q. Let , :Q → TQ be a sections. Then, the covariant derivative of ∘ along  is     
    (∘) =  ∇^_() + D^(),
 where the vertical Jacobian D^ = _D |_ TQ is given locally by 

    D^() = ^mw^l()^l q^m,
 and the horizontal connection () = _D|_ TQ is given locally by 

    () = (^mq^kv^k  + Γ^m_kiv^k^i - Γ^l_kiw^kv^i^mw^l)q^m.



    See appendix <ref>




    Note that when the bundle map :TQ → TQ linear in velocities, the vertical derivative D^ = is linear in its argument, and the horizontal covariant derivative  reduces to the usual covariant derivative  e.g., see equation (<ref>). Therefore, this "chain rule" is a generalization of the traditional chain rule for the covariant derivative of a linear vector bundle map. 



    For practical calculations, in the coordinate induced basis {q^1,…,q^n}, the covariant derivative  given in (<ref>) may be calculated in matrix form as
    
    = [] + [(,)],
 where [] = [Y^iq^k] is the Jacobian of  and the Christoffel symbols are organized in a matrix [(,)] = [Γ^i_kjX^j]. Similarly, the covariant derivative of a (1,1)-tensor  given in (<ref>) may be calculated in matrix form as
    
    [] = [∑^n_k=1 q^kX^k] + [(,)][] - [][(,)].
 Lastly, the covariant derivative  of a tangent bundle map (,) as in (<ref>) may be calculated in matrix form by 
    
    = [(,)] + [(,)] - [D^][(,)].





  §.§.§ Dynamics of Mechanical Systems on TTQ

A vector field :TQ → TTQ is a second order system on Q if and only if for each integral curve :[a,b]→ TQ of  we have t(π_TQ∘)(t) =  (π_TQ)_* = (t), summarized in the following diagram <cit.>




    TTQ 
    
    	TQ        TQ       
        Q
    	["(π_TQ)_*"'pos=0.2, from=1-2, to=2-1]
    	["π_TTQ"pos=0.3, from=1-2, to=2-3]
    	["π_TQ", from=2-3, to=3-2]
    	["π_TQ"', from=2-1, to=3-2]
    	["X"', curve=height=30pt, from=2-3, to=1-2]
    	["X", curve=height=-30pt, from=2-1, to=1-2]
    	["id_TQ"', from=2-1, to=2-3]


 
If = (,,,) is a second-order vector field, then it follows that the velocity components satisfy v^i = V^i. 

Given, the decomposition TTQ ≅ TQ ⊕ TQ, we construct the vector field associated with the affine connection system (<ref>). Let (,):[a,b] → TQ be a curve of satisfying (<ref>). The forces acting on the system (- V^♯_ +F^♯_) in the right-hand side of (<ref>) are vertically lifted to _(,)TQ by (<ref>). The vector field associated with the parallel transport equation (t)(t) = on the left-hand side of (<ref>) is given by the horizontal lift (<ref>) of the velocity  to _(,_)TQ.
Therefore, the time variation of the curve t(,) ∈ T_(,)TQ ≅_(,)TQ⊕_(,)TQ satisfies the differential equations 


    t = -Γ^k_ijv^iv^jv^k + ( - Vq^i + F_i)^ikv^k,
       t = v^iq^i.
 

    When there are no external forces in (<ref>), the vector field :TQ → TTQ
    
    (,) = v^iq^i - Γ^k_ijv^iv^jv^k,
 is called the geodesic spray <cit.>. The integral curves of the geodesic spray  in TQ project to geodesics with respect to  in Q.










§ AFFINE CONNECTION SYSTEM AS A SLOW/FAST SYSTEM WITH STRONG FRICTION
  

Let Q be a smooth manifold and let ⊂ TQ be a linear subbundle representing a set of m ideal nonholonomic constraints. To realize the nonholonomic constraints by friction forces, consider a Rayleigh dissipation function :T_qQ× T_qQ →, represented as a symmetric (0,2)-tensor on each tangent space <cit.>. This function induces a dissipative friction force given by the (negative) fibre derivative -:TQ → T^*Q of the dissipation function. Assume that this friction force satisfies = () to ensure that the nonholonomic distribution is attractive. 

We construct this dissipation function   as follows. Consider a ^m-valued one-form a_:T_Q →^m consisting of the m nonholonomic constraint one-forms a^l∈ T^*Q for l = 1,…,m. For each configuration ∈ Q, define the friction coefficients as the symmetric linear vector bundle map μ̅(): ^m→^m*≅^m and finally, define the adjoint of the map representing the constraint as a^*_:^m*→ T^*_Q. Define the Rayleigh dissipation function = 1/2a^*_∘μ(q) ∘ a_ and in local coordinates is given by 

    = 1/2_i^l()μ̅_lk()^k_j()__ij() q^j⊗ q^i.

The dissipative Rayleigh friction force with large coefficients is defined as follows. Fix a small positive real number >0 (which is thought to be the time scale on which this force acts) and rescale the friction coefficients μ̅ = 1/μ to find the friction force given by 

    F_ = -1/() = - 1/_ij()v^i q^j∈ T^*Q.
 Let :[a,b] → Q be a smooth curve that satisfies the equations of motion formulated by the forced affine connection system 

    = -( V )^♯_  - 1/()^♯_,
       t = .



 As this is a singularly perturbed system, it follows that (<ref>-<ref> emits a normally hyperbolic invariant manifold (NHIM) for >0 <cit.>. That is, for small values of , there exists a perturbed manifold _ that is C^1-close and diffeomorphic to the nonholonomic constraint distribution . The existence of this submanifold has been studied by Eldering and summarize the result <cit.> as follows. Write the system as a vector field (<ref>) on TQ according to the decomposition TTQ ≅ TQ ⊕ TQ: 
 
    t(,) = (,) + (- V^♯_)^ - (1/()^♯_)^
 where we have vertically lifted the forces to TTQ using (<ref>). Make the reparameterization of the trajectory to fast-time τ = t/ to find  
 
    τ(,) = (,) +(- V^♯_+ U^♯_)^ -  (()^♯_)^.
 When = 0 then the fast time system is of the form 
 
    τ(,) = - (()^♯_)^.
 When this friction force acts on an instantaneous time-scale (= 0) it is the dominant force acting on the system. By construction, when we restrict the dynamics to the distribution , the friction force vanishes; thus  is an invariant manifold for the fast-time dynamics (<ref>). When the dynamics are restricted to the -orthogonal complement ^, we find that the dynamics are attractive. Therefore, by the theory of normally hyperbolic invariant manifolds <cit.> there exists a perturbed manifold _ that is C^1-close and diffeomorphic to .

 

 §.§ Representation of the Slow Manifold

 We represent the perturbed manifold _ as the image of a section _:→ TQ over the bundle :TQ →. By definition, the section _ satisfies the following commutative diagram














    𝒟       TQ 
       𝒟["H̃_ε", from=1-1, to=1-3]
    	["𝒫", from=1-3, to=2-2]
    	["id_𝒟"', from=1-1, to=2-2]


 That is, ∘_ = id_. By the projection property id_TQ = ⊕^, we find that the section _ is given by 
 
    _ = id_ + ^∘__=:_
 where _:→^ is a section over π_TQ:TQ → Q. Therefore, the slow manifold _ is defined by the set 
 
    _ = {∈ TQ   |   = ^ + _(^),  ^∈}.

 
 
 Using the projection maps, we then define the sections _ = _∘:TQ→ TQ as 
 
    _() = () + _∘(),
 where _∘ = ^∘_∘:TQ →^. Therefore, the perturbed manifold _ = (_) is given by 
 
    _ = {∈ TQ  |   = _() = () + _∘(),  ∈ TQ }.
 Note that the form of this function _ depends on the choice of projection map . 
 
     Physically speaking, the slow manifold represents the slip velocities in the orthogonal complement ^ and are a function of the variables in the nonholonomic distribution . 
 


    The map _:TQ → TQ for the system (<ref>)-(<ref>) is a nonlinear projection onto _. 


    Given a section :Q → TQ we find
     
    _∘_()    = _(() + _∘() )
       = (_∘)(() + _∘() )
       = _(∘() + ∘_∘() )
       = _(∘() + ∘^∘_∘() )
        = _∘() = _().

  

    Since _ defines a nonlinear projection map, we may also define its complement ^_ := id_Q - _. This defines a projection map onto the complement of the slow manifold abbreviated by TQ/_. 

 The map ^_:TQ→ TQ/_ is a nonlinear projection map that is complementary to _.


    First, we show that _ is a projection map. Let :Q→ TQ be a section. Then, 
    
    _^∘_^()   =(id_TQ - _)∘( - _())
       = - _() - _() + _∘_()
       =  - _() = _^().
 Lastly, we show that this map is complement to _.
    
    _∘_^() = _() - _∘_() = 0,
       _^∘_() = _∘_() - _()   = 0.



    As a consequence of the complementary projection map in Proposition <ref>, we find that ∈_ if and only if 
    
    _^()    =  -_() =  - () - _∘()
       = ^() - _∘() = 0.
 This is summarized in the following proposition.  


  
     ∈_ if and only if ^() = _∘().
 
  Suppose that ∈_. Then, there exists ∈ TQ such that = () + _(). We calculate
     
    ^()    = ^(() + ^∘_())
       = ^∘^∘_() = _() = ^∘_∘().
 Moreover, we find that  and  project to the same vector
 
    () = (() + _∘() )) = ().
 Therefore, ^() = ^∘_∘() = _∘().
 
 Conversely, suppose ∈ TQ such that ^() = _∘(). Then, since  is a projection map, we have that _TQ = ⊕^. Therefore, since we assume that ^() = _∘(), we find
 
    = () + ^() = () + _∘().
 Therefore, ∈_.
  By Proposition <ref>, we find that the distribution can be represented as the set

    _ = {∈ TQ   |  ^() = _∘() }.

 

    The equations of motion for a curve :[a,b] → Q lying on the slow manifold _ are given by

    (t)(t) = - V^♯_  - 1/((t))^♯_
       t(t) = (t),
       ^((t)) = _∘((t)) (t) ∈_.
 


    The equations of motion (<ref>)-(<ref>) are described by a forced affine connection system. The existence of the slow manifold follows from the geometric singular perturbation theory of normally hyperbolic manifolds <cit.>. By Proposition <ref> the slow manifold is represented by (<ref>).

















§ APPROXIMATION OF THE SLOW SUBSYSTEM
 

     In this section, we find a generating equation for the invariant manifold _ using the equations of motion (<ref>) - (<ref>).  The strategy is to consider velocities restricted to the slow manifold _ and expand the section _ as a power series and match terms to the desired order in ϵ. 



 §.§ Generating equation for the slow-manifold

Let us find an expression for the dissipation function ()^♯_ in terms of the unknown function _ describing the perturbation of the ideal constraints. Using the projection maps, we project the dynamical equations (<ref>) onto ⊕^≅ TQ

    () = -( V^♯_)
       ^() = -^( V^♯_) - 1/^♯_().
 


Consider the transformation ^♯__:T_Q → T_Q, then there exists a transformation _:^_→^_ satisfying _∘^♯__ = ^_.
 

    The friction force is a linear transformation ^♯__:T_Q → T_Q satisfying = (^♯__) and im(^♯__) = ^. Thus, by the first isomorphism theorem <cit.>, we have the result.











 
 
    The map _ may be constructed as follows. Let Φ() be a frame spanning _⊕^_. Define linear maps in this frame with the subscript Φ, so, [_Φ] = Φ^-1[]Φ, [^_Φ] = Φ^-1[^]Φ and [_Φ^♯_] = Φ^-1[^♯_]Φ and the equation [_Φ^]=[_Φ][_Φ^♯_] is given by the block matrix equation of the form
    
    [  ;   ]_[^_]  = [  ;   ]_[_Φ][  ;   ]_[_Φ^♯_].
 Therefore, we set = ^-1 and perform the inverse transformation [_] = Φ[_Φ]Φ^-1 to obtain the result.


 
    Let _:→ TQ be the section defining the invariant manifold _ with respect to the dynamics (<ref>)-(<ref>). Let _ = ∑_k ≥ 0^k^(k) be an expansion of the section by a formal power series. Then, the slow manifold may be recursively approximated from the following generating equation 
    
    _(^) = _∘[(^^)(^) - ^( V^♯_) ] 
        + _∘[ (^^)(_) + (_^)(^) + (_^)(_) ] 
       - _∘ D^_[(^)(^)  - ( V^♯_) ] 
        - _∘ D^_[(_)(^)  + (^)(_) + (_)(_)  ] 
              - _∘[__(^) + __(^) ].




    By analogy with the ideal nonholonomic case <cit.>, take the covariant derivative of the latter invariant manifold equation (<ref>) 

    (^()) = (_∘())
 and substitute the affine connection system (<ref>)-(<ref>). The left-hand side is given by 

    (^() )   = (_^)() + ^(_)
       = (_^)()  -1/()^♯_ - ^( V^♯_).
 On the other hand, according to Proposition <ref> and the chain rule, we find the right-hand side 

    (_∘() )= D^_((()))  + _(())
        = D^_(()() + () )  + _(())
        = D^_(()() )-D^_(( V^♯_))  + _(()).
 Putting both sides of this calculation together, we find the following identity for the dissipative forces 

    -1/^♯_() = -(^)() + ^( V^♯_)
       + D^_(()() ) + _(()) - D^_(( V^♯_))
  

Lastly, we restrict (<ref>) to the velocities ∈_ to obtain the generating equation for _

    ^♯__(^)   = (_^ + _^)(^ + _) + ^( V^♯_)
        -  D^_(- V^♯_)  - (_^ + __)(^) 
        + D^_( (^ + _)(^ + _) ).
 By Lemma <ref> and linearity we obtain the generating equation (<ref>).


    Note that there is an  in all terms on the right-hand side of (<ref>). Thus, any power series expansion of _ results in an order higher on the right-hand side than the left-hand side. Therefore, the generating equation (<ref>) provides a recurrence relation for the approximation of _.

 
  The generating equation for _(^) in (<ref>) results in the same reaction forces -1/^♯__(^) as those reaction forces resulting from the classical invariance condition 

    t^() |__ = t_∘() |__.



    Without loss of generality, assume that the potential V=0. The equations of motion (<ref>)-(<ref>) can be written as 
    
    = -(,) - 1/^♯_()
       =∈_^() = _∘().
 Taking a Lie derivative of (<ref>) along the along these dynamics we find 
    
    [∑^n_k=1^q^kv^k] - [^][(,)] - 1/[^][^♯_] = 
       [_(,())] + [D^_][∑^n_k=1q^kv^k]
          - [D^_][][(,)] - 1/[D^_][][^♯_].
 where the quantities in the square brackets are the compact matrix notation introduced in Remark <ref>. As = ^♯_, the last term vanishes. Let = ^_∈_ be a velocity in the slow manifold, then the reaction forces are restricted to _ are given by 
    
    -1/^♯_(^_) = -[∑^n_k=1^q^kv^k__]^_ +[^][(,^_)]^_
            +[_(,^)]^_ + [D^_][∑^n_k=1q^kv^k__]^_- [D^_][] [(,^_)]^_.
 Contrastingly, the reaction forces _ resulting from the generating equation (<ref>) are calculated from the equation
    
    -1/^♯_() = -(^)() + D^_()() + _(()).
 as was observed in equation (<ref>). By Remark <ref> the covariant derivatives in this equation may be expressed in matrix form by 
    
    -1/^♯_() = -[∑^n_k=1^q^kv^k] - [(,)][^] + [^][(,)]
        + [D^_][∑^n_k=1q^kv^k] + D^_[(,)][] -[D^_][][(,)]
        + [_(,^)] + [(,)]_(,^)- [D^_][(,)][].
 Then, restricting to = ^_, and using ^∘_ = _ terms cancel and therefore, these friction forces (<ref>) restricted to _ are identical to (<ref>).


 The dynamics defined by (<ref>)-(<ref>) restricted to _ are tangent to the manifold _.


    In local coordinates (<ref>)-(<ref>) are equivalent to the equations of motion on TTQ given by (<ref>). Following singular perturbation theory, the classical invariance equation (<ref>) ensures that the reaction forces result in dynamics tangent to _.
    By Theorem <ref>, the reaction friction forces described by (<ref>) are equivalent to the reaction forces resulting from the classical invariance equation. Therefore, the dynamics (<ref>)-(<ref>) along with the generating equation (<ref>) result in dynamics tangent to _. 



    The classical invariance condition (<ref>) for the dynamics (<ref>)-(<ref>)
is equivalent to the condition 

    (^())|__ = (_∘())|__.



    By Theorem <ref> and Corollary <ref> both conditions result in the same dynamical equations of motion restricted to the slow manifold _. 


    The later covariant derivative invariance condition (<ref>) may be geometrically interpreted as the parallel transport of the condition ^() = _∘() along trajectories = ∈_ lying in the slow manifold.






 §.§ Approximation of Slip Velocities

In this section, we employ the generating equation (<ref>) from Theorem <ref> to compute the zeroth, first and second order approximations of the slow manifold _. Expand _ as a formal power series 

    _ = ^(0) + ^(1) + ^2^(2) +  O(^3).
 Further expanding the power series allows for higher order approximations of _.


  §.§.§ Zeroth Order Approximation of Slip Velocities
 Lets consider the zeroth order term of the slow manifold _ = ^(0) + O(). Observe that when = 0 we have 

    _ |_ = 0 = _0  = {∈ TQ   |  ^() = ^(0)}.
 Since _ is a perturbation of the nonholonomic distribution, we find that ^(0) =. Then, to find the zeroth order velocity,  solve the identity ^() = for = ^∈.

    The zeroth order approximation of the slow fast velocities are equivalent to the nonholonomic velocities.  





  §.§.§ First Order Approximation of Slip Velocities
 Let us assume that section _ is taken to first order, so we expand _ = ^(1) + O(^2). Then according to the generating equation (<ref>) to this first order slip velocity, we find 

    ^(1) = _∘[(^^)(^) - ^( V^♯_)].



    The first order approximation of the slow fast velocities are proportional to the nonholonomic Lagrange multipliers (<ref>). This is consistent with previous results <cit.>.





  §.§.§ Second Order Approximation of Slip Velocities
 Lastly, assume that section _ is taken to second order, so we expand the formal power series as _ = ^(1) + ^2^(2) + O(^3). Then according to the generating equation (<ref>) to this first order slip velocity, we find 

    ^(2) = _∘[(^(1)^)(^) + (^^)(^(1))] 
              -_∘[D^^(1)((^)(^) ) - D^^(1) ( V^♯_) ]- _∘[^^(1)(^) ].



    Higher order approximations of _ may be calculated recursively in terms of higher order covariant derivatives ^n. However, in practice, first and second order approximations suffice. This is due to the fact that the friction forces - 1/^♯_() in (<ref>) restricted to the second order approximation results in a first order force - ^♯_(^(2)) that is a function of zeroth and first-order velocities. 





 §.§ Approximation of Slip Dynamics

In this section, we use the approximations of the slip velocities _ = ^(1) + ^(2) + O() to approximate the zeroth and first order dynamics. Due to the recurrence relation defined by the generating equation (<ref>) on _ we observe that the zeroth order approximation of the dynamics (<ref>)-(<ref>) depends on the first order approximation ^(1). Furthermore, the first order approximation of the dynamics (<ref>)-(<ref>) depends on the second order approximation ^(2), and so on. Therefore, the order of the slip velocities _ is always one order above the order of the dynamics. 



  §.§.§ Zeroth Order Dynamics
 By the equations of motion (<ref>) and the first order approximation of the slow manifold _ = ^(1) + O(^2), we find the zeroth order equations of motion are given by

    t = -Γ(,^)^ - (^^)(^) - ( V^♯_)
       t  = ^∈_0≅.



    The zeroth order equations of motion (<ref>) are exactly the nonholonomic equations of motion (<ref>)-(<ref>). This result is consistent with previous realizations of slow/fast nonholonomic systems <cit.>. 





  §.§.§ First Order Dynamics
 Using the generating function (<ref>) we obtain the first and second order approximations _ = ^(1) + ^2^(2) + O(^3) to obtain (<ref>) and (<ref>), the former gives the slip velocity and the later gives the reaction force. Restricting the dynamics to the first order approximation of slip, we find the dynamics: 


    t   = 
        -Γ(,^)^ - (^^)(^) - ( V^♯_)
          -[Γ(,^)^(1) +Γ(,^(1))^] +^^(1)(^)
          -[(^^)(^(1)) + (^(1)^)(^) ]
           +   D^^(1)[((^)(^) ) - ( V^♯_) ]
        
       t = ^ + ^(1)(^)
 where the first order approximation of the slip ^(1) is given by (<ref>), and vertical derivative D^^(1) and horizontal ^^(1) is calculated according to equation (<ref>) and (<ref>). 



    











§ CASE STUDY: VERTICAL ROLLING DISK
 
To illustrate the theory, we consider a vertical rolling disk for simplicity. However, our approach may be applied to other systems with (linear) nonholonomic rolling constraints such as wheeled mobile robots <cit.> and vehicle systems <cit.>. For the traditional derivation of the nonholonomic equations for the vertical rolling disk using the Lagrange-d'Alembert variational principle see <cit.> and for the affine-connection approach see <cit.>, and <cit.>. 

The configuration space for a planar vertical rolling disk is Q = SE(2)× SO(2) with local coordinates = (θ,x,y,φ) and local coordinates for velocities = (v_θ,v_x,v_y,v_φ). The Riemannian metric defining the kinetic energy is given by 

    _ = diag(I,m,m,J )
 where m is the mass of the disk, I is the moment of inertia normal to the plane and J is the moment of inertia about the rolling axis. Note that as _ is constant, the corresponding Christoffel symbols (,) = for all (,) ∈ TQ , the metric _ is flat. 

The nonholonomic constraint one-forms 

    a_q^1 =  x - Rcosθφ,   a_q^2 =  y - Rsinθφ,
 enforcing rolling without slipping are given by 

    a_q^1(v_θ,v_x,v_y,v_φ) = v_x - Rcosθ v_φ = 0,
       a_q^2(v_θ,v_x,v_y,v_φ) = v_y - Rsinθ v_φ = 0,
 where R is the radius of the disk. This constraint defines a nonholonomic distribution _ = (a^1_q)∩(a^2_q) which are spanned by the vector fields 

    θ,    Rcosθx + Rsinθy + φ.
 Using the musical isomorphisms, we find that the -orthogonal complement spanned by the orthogonal complement ^ = (^∘)^♭_ is spanned by the vector fields 

    1/mx -R/Jcosθφ,   1/my - R/Jsinθφ.
 The projection map :TQ → onto the distribution  is constructed by the vector fields spanned by (<ref>) and the metric (<ref>) and is given by 

    = [       1/        0        0        0;        0   cos^2θ 1/2sin2θ J/mRcosθ;        0 1/2sin2θ   sin^2θ J/mRsinθ;        0   cosθ/R   sinθ/R   J/mR^2 ]
 where := mR^2/J+mR^2 is a ratio of inertia. 

Similarly, the projection map ^:TQ →^ onto the distribution ^ is constructed by the vector fields spanned by (<ref>) and the metric (<ref>) and is given by 

    ^ = [                   0                   0                   0                   0;                   0 J + mR^2sin^2θ/mR^2           -1/2sin2θ           -J/mRcosθ;                   0           -1/2sin2θ J + mR^2cos^2θ/mR^2           -J/mRsinθ;                   0             -cosθ/R             -sinθ/R                   1 ].
 The nonholonomic constraints may then be expressed by the equation ^() =, from which we find the constrained velocities ^∈ is given by 

    ^ = [       v_θ Rcosθ v_φ Rsinθ v_φ       v_φ ]^T
 in the coordinate induced frame {θ,x,y,φ}.

We construct the Rayleigh dissipation function according to equations (<ref>) using the nonholonomic constraint form (<ref>), from which the coefficient matrix is given by 

    ^♯_ = [          0          0          0          0;          0        μ/m          0 -μ R/mcosθ;          0          0        μ/m -μ R/msinθ;          0 -μ R/Jcosθ -μ R/Jsinθ    μ R^2/J ]
 where the units of the friction coefficients μ are [kg/s ], and the corresponding accelerations given by _ = -1/^♯_. Defining a frame Φ as the direct sum of the vector fields (<ref>) and (<ref>) we find the matrix _ according to Remark <ref> is given by 

    _ = [                    0                    0                    0                    0;                    0 J^2 + γ m R^2 sin^2θ      -1/2γ mR^2sin2θ             J^2Rcosθ;                    0      -1/2γ mR^2sin2θ J^2 + γ m R^2 cos^2θ          - J^2 Rsinθ;                    0             -mRJcosθ             -mRJsinθ                mR^2J ]
 where γ = mR^2 + 2J. One can check that _ satisfies Lemma <ref>. As the Christoffel symbols are all zero, the equations of motion are simply given by 


    t[ v_θ; v_x; v_y; v_φ ]  = -1/[                                        0;                    μ/mv_v - μρ/mcosθ v_φ;                    μ/mv_y - μρ/msinθ v_φ; -μρ/Jcosθ v_x - μρ/Jsinθ v_y + μρ^2/Jv_φ ],   
       t[ θ; x; y; φ ] = [ v_θ; v_x; v_y; v_φ ].


    Note that the dynamics of the orientation in the plane θ is of the form v̇_θ = 0 as v_θ∈ and = (^♯_), by construction. Hence, the orientation in the plane is invariant for the dynamics of the vertical rolling disk. 




 §.§ Approximations of Slip Velocities
 The first order approximation of the slip velocities are calculated according to  (<ref>) and are found as 

    ^(1) = _∘(^^)(^) =  [                0;  mR/μsinθ v_θv_φ; -mR/μcosθ v_θv_φ;                0 ]
 Further, the second order approximation of the slip velocities is calculated according to (<ref>) using the results of Remark <ref> and is found to be 

    ^(2)   =     _∘[(^(1)^)(^) + (^^)(^(1))] 
              -_∘[D^^(1)((^)(^) ) ] - _∘[^^(1)(^) ] 
       =m^2RJ^2/μ^2(J + mR^2)^2[              0; cosθ v_θ^2 v_φ; sinθ v_θ^2 v_φ; -mR/Jv_θ^2 v_φ ]
 




 §.§ Approximations of Slip Dynamics
 Using the first order approximation of the slip velocities (<ref>), we find the zeroth order approximation of (<ref>)-(<ref>) using equations (<ref>) are given by 

    t[ v_θ; v_x; v_y; v_φ ] = [              0; -Rsinθ v_θ v_φ;  Rcosθ v_θ v_ϕ;              0 ] +O(), 
       t[ θ; x; y; φ ] = [       v_θ; Rcosθ v_φ; Rsinθ v_ϕ;       v_φ ] +O().
 


    The zeroth order equations of motion (<ref>)-(<ref>) are the same equations of motion resulting from Lagrange-d'Alembert's variational principle <cit.>. 

Moreover, using the second order approximation of the slip velocities (<ref>) and (<ref>), we find the first order equations (<ref>) are given by 

    t[ v_θ; v_x; v_y; v_φ ] = [              0; -Rsinθ v_θ v_φ;  Rcosθ v_θ v_φ;              0 ] +mRJ/μ(J+mR^2)[              0; -cosθ v_φv_θ^2; -sinθ v_φv_θ^2;  mR/J v_φv_θ^2 ]+O(^2),
       t[ θ; x; y; φ ] = [       v_θ; Rcosθ v_φ; Rsinθ v_φ;       v_φ ] + mR/μ[            0;  sinθ v_θv_φ; -cosθ v_θv_φ;            0 ] +O(^2).
 




§ CONCLUSION

In this paper, we developed an affine connection approach for realizing nonholonomic mechanical systems. We formulated the slow/fast dynamics in a coordinate-independent manner in terms of sections, projection maps and covariant derivatives. We proposed a novel invariance condition for slow/fast realizations of nonholonomic constraints and proved that this novel condition is equivalent to the classical approach. We proposed a recursive procedure to approximate higher order slip velocities and their associated dynamics. We conclude the paper with some remarks regarding the advantages of the proposed formalism. 

    
  * The geometric formalism is coordinate-independent and suitable for symbolic computation.
    
  * The proposed slow-fast decomposition permits a recursive procedure to approximate the invariant manifold representing slip velocities. In the context of vehicle systems, the slow manifold may be physically interpreted as a deviation from the ideal no-slip constraints. Approximations of the slow manifold provide estimates of these deviations and have applications in model-based slip estimation and localization in vehicle systems. 
    
  * Higher order approximations of slip velocities are based on objects that can be interpreted geometrically and may give insight into the geometric structure of other equivalent approaches. 
 The future directions of this research include the following: (i) Designing a recursive controller robust to the uncertainties in the estimation of the friction coefficients and external forces; (ii) Use of higher-order approximations of the slow-manifold for in-the-loop motion estimation of vehicle systems; and (iii) symmetry reduction for slow-fast realizations of nonholonomic systems. 







Conflict of Interest:  The authors declare that they have no conflict of interest.





§ PROOFS OF STATEMENTS




 §.§ Proof of Proposition <ref>
 
 
    For the horizontal space  

    _(,_,,)    = (_Γ^(,_,,))^
       = (,_,,- Γ(,_))^
       = (,) = (π_TQ)_*(,_,,).
 Therefore we have _ = (π_TQ)_*. 

For the vertical space, let :Q → TQ be a section:

    _(_*)    = (^_Γ(,_,_,D()·_) )^
       = (,_,, D()·_ + Γ(,_)·_)^
       = (,D()·_ + Γ(,_)·_)
        = .







 §.§ Proof of Proposition <ref>
 












Let :TQ → TQ be a vector bundle map, i.e., π_TQ∘ = π_TQ. Let , :Q→ TQ be sections, then ∘ : Q → TQ is also a section. The push-forward of this section is a map that satisfies the following commutative diagram: 




    TTQ        TTQ 
    
    	TQ        TQ 
        Q
    	["h", from=2-1, to=2-3]
    	["π_TQ", from=2-3, to=3-2]
    	["π_TQ"', from=2-1, to=3-2]
    	["w", curve=height=-24pt, from=3-2, to=2-1]
    	["π_TTQ", from=1-3, to=2-3]
    	["h_*", from=1-1, to=1-3]
    	["w_*", curve=height=-12pt, from=2-1, to=1-1]
    	["π_TTQ", curve=height=-6pt, from=1-1, to=2-1]


 By virtue of Proposition (<ref>), we calculate the covariant derivative along  as follows

    (∘)   = _(D(∘) ) = _(D· D)
       = _(D· (_⊕_)(D)  )
       = _(D·(_(D)+_(D)  ))
       = _(D·(_(D)⊕) ) + _(D·(⊕) ),
 where we used the direct product isomorphism, twice and D is the Jacobian of . In the decomposition TTQ ≅ TQ⊕ TQ, consider the Jacobian of the map :

    D = [ _D|_ _D|_; _D|_ _D|_ ].
 Since _ = (π_TQ)_*,  when restrict D to the vertical space TQ = (π_TQ)_* we find 

    π_D|_ = (π_TQ)_*_*|_  = (π_TQ)_*|_  = .
 Let :[a,b] → TQ be a horizontal curve along (t) in Q. Then, ∈ T_(t)TQ ≅_(t)TQ⊕_(t)TQ satisfying 
    = ^i(q^i - Γ^k_ijv^jv^i).
 Since ((t)) ∈ T_(t)Q is a curve, we calculate derivative 

    (t)    = t(π_TQ∘∘(t)  ) = (π_TQ)_*  D  = _(D ).
 Therefore, we find that _D|_ is the identity map id. 

Define the vertical component of the Jacobian by D^:= _(D)|_ and the horizontal covariant derivative ∇^:= _D|_. 
Therefore, the covariant derivative of the bundle map  is given by 

    (∘) = () + D^().

We calculate local coordinate expressions for the covariant derivative as follows:

    (∘)   = v^iq^i((∘)^jq^j)
       = v^i (_q^i((,)^j)q^j + (,)^jq^iq^j)
       = v^i (^jq^iq^j + ^jw^kw^kq^iq^j + ^iΓ^k_ijq^k).
 On the other hand we have that 

    q^i = (w^kq^i + Γ_ijw^j)q^k.
 Putting this expression into the above formula we find 

    (∘)   = ^mw^l()^l q^m + (^mq^kv^k  + Γ^m_kiv^k^i - Γ^l_kiw^kv^i^mw^l)q^m
       =  D^() + ().
 




















 Paper Title: Non degenerate Birkhoff functions.


§ ABSTRACT   It is known after the works of Mahler, Kleinbock, Margulis, Sprindžuk that very well approximated numbers on a manifold form a zero measure set, assuming non-degeneracy conditions.  These non-degeneracy conditions are,
  in many applications, difficult to check. We propose here a setting in which these are automatic. Rather than functions, we consider correspondences which have a better behaviour.

Interpretable correlator Transformer for image-like quantum matter data
    Annabelle Bohrdt
    August 5, 2024
=======================================================================



 


§ INTRODUCTION

In 1942, Siegel proved that a biholomorphic germ

    f:(,0) → (,0),  z ↦ z +o(|z|)

can be linearised, assuming an arithmetic condition on the coefficient
 <cit.>. Ever since, the field of Diophantine
approximation has been of relevance for dynamical systems and this led
Arnold, Margulis and Pyartli and others to reconsider the classical works
on Diophantine approximation and extend their scope <cit.>. Nowadays, the work of Kleinbock and Margulis finds application
in KAM theory in order to prove that in certain situations the quasiperiodic
solutions make up a set of positive measure in phase space <cit.>.
To apply these results, one needs to assume certain non-degeneracy conditions, introduced in the works of Mahler, Sprindžuk, Margulis and Kleinbock <cit.>. To our knowledge the weakest of these conditions was introduced by Kleinbock:
 Let U be an open set in ^d. A C^k-map B:U →^e is called KMS-non-degenerate (for Kleinbock-Margulis-Sprindžuk) in an affine space  if:

 
  * B(U) ⊂.
 
  * For some l ≤ k the partial derivatives at any point of U of order l span the vector space associated to . 


It is not difficult to see that real analytic maps are always KMS non-degenerate (see <ref>) and this will constitute our starting point. A simple example of a KMS-degenerate function is given by the flat function:

    →, x ↦ e^-1/x^2

 
 
 Assuming the KMS non-degeneracy condition, one may estimate the preimage under the map B of points which are badly approximated by the integer lattice ^e ⊂^e in various ways. For instance, under the KMS non-degeneracy condition, Kleinbock-Margulis and Kleinbock proved that the preimages of points satisfying an adequate Diophantine condition form a full measure set (<cit.>, <cit.>, see also <cit.>). 
  
Unfortunately the verification of non-degeneracy conditions for the frequency map of a dynamical system often rely on tedious computations and are notoriously difficult to establish. For instance, it took  many years before Féjoz and Herman proposed a proof of the non-degeneracy condition for the N-body problem initiated by Arnol'd in the sixties (<cit.>, see also <cit.>). In this paper, we will show that for the type of normal forms introduced by the authors in <cit.> the KMS-non degeneracy conditions are essentially automatic. So let us now explain what type of generalisation of the usual setting is needed by considering an instructive example.


 §.§ The Poincaré correspondences




In Les Méthodes Nouvelles de la Mécanique Céleste <cit.>. 
Poincaré considered the series


    s=1+x/1+y+x^2/1+2y+…=∑_k=0^∞x^k/1+k y

as a prototype of the kind of series that appear in perturbation theory (see also the paper <cit.> of Sauzin for the resurgence properties). In fact, similar expressions play a role in many areas of mathematics, for example in the theory of q-analogs of classical series (logarithm, exponential, hypergeometric), see for instance <cit.>.

Assume now that we want to solve the equation

    0=f(x,y):=-y+x/1+y+x^2/1+2 y+ +x^3/1+3 y+… .


If we truncate the series at x^n we obtain the rational functions

    f_n(x,y)=-y+∑_k=1^n x^k/1+ky =P_n(x,y)/D_n(y)

which determines a polynomial  P_n(x,y) in the variables x and y:

    P_1(x,y)=-(1+y)y+x,   P_2(x,y)=-(1+y)(1+2y)y+(1+2y)x+x^2, …


    D_1(y)=1+y,   D_2(y)=(1+y)(1+2y),    D_3(y)=(1+y)(1+2y)(1+3y),…

These polynomials define algebraic curves

    V_n:={ (x,y) ∈^2:P_n(x,y)=0 }⊂×

of degree (n+1) in y and degree n in x. These curves exhibit a peculiar bending behaviour caused by the poles, located on horizontal lines, defined by the denominators D_n(y) that were cleared out. 


  
Such a polynomial relation defines y as a multivalued function of x, sometimes called a correspondence. In a neighbourhood of x=0,y=0, the n-th curve V_n is the graph of a holomorphic function b_n that can be expanded in power series:

    [                  b_3(x)                       =    x+2x^5-14x^6+64x^7+…;                  b_4(x)                       = x+x^4-3x^5+7x^6-19x^7+…;                  b_5(x)                       =  x+x^4-2x^5+x^6+12x^7+…;                  b_6(x)                       =  x+x^4-2x^5+2x^6+5x^7+…;                         ]

It is not difficult to see that b_n+1(x)=b_n(x)+o(x^n), so these series converge coefficientwise to a formal power series solution b(x) ∈[[x]] to the equation f(x,y)=0

    b(x)=x+x^4-2x^5+2x^6+6x^7-35x^8+86x^9+…

We call it the Birkhoff expansion of the series f and its truncation are called the Birkhoff polynomials of the series.

 
 

 



The domain of definition of the b_n shrinks as n →∞ due to the peculiar winding, which provides a geometrical reason for the divergence of the series b(x). This is an illustration of a general divergence theorem for such expressions, that we intend to publish elsewhere <cit.>. On the other hand, the pictures strongly suggest that the part of curves in the part y>0 nicely converge to the graph of a C^∞-function B.


We may construct such a C^∞ function as follows: start with closed disks

    D_ρ:={x ∈: |x| ≤ρ}, D_η:={y ∈: |y| ≤η},

where ρ<1  and remove from the disk D_η small disks
D_k:={ y ∈: |y+1/k| < r_k } around the poles y=-1/k.
The distance between two nearby poles being 1/k(k+1),
 if we choose r_k=k^-, >2, the complement of the disks D_k contains the origin. We have in the complement of D_k an obvious estimate:
 
    |x^k/1+ky|≤ k^-1|x^k| ≤ k^-1ρ^k

So for  ρ <1, we see that
 the sequence obtained by restricting the rational functions f_n(x,y) to the closed set 
 
    X_∞ :=D_ρ×( D_η∖∪_k D_k )

converges in the C^0-topology to a limit function, that we denote in the same way: 

    f: X_∞





 §.§ Whitney differentiability

The closed set X_∞ does contain 0, but it is not an interior point as the poles accumulate at the origin. However, the formal power series b  that solves the equation f(x,b(x))=0 is the Taylor expansion of a function, C^∞ in a sense originally discovered by Borel and now called Whitney differentiability <cit.> Let us explain this property.  

If X ⊂^k is a locally closed subset, then a function 

    g:X →
 is called Whitney differentiable if  there exists a map

    L:X →  Hom(^k,)

such that

    g(y)=g(x)+L(x)(y-x)+o( x-y) .

The main difference with usual derivatives is the requirement that the variables x,y play a more
symmetric role. One defines similarly higher derivatives and Whitney C^∞ functions. Holomorphicity in the Whitney setting is defined by requiring that the map L is -linear
(^d=^2d) and in that case the function was called monogenic by Borel.

The Whitney extension theorem states that Whitney differentiable functions are restrictions of usual differentiable functions and the Whitney derivative commutes with restriction <cit.>. Fefferman showed that the extension operator can be chosen to be bounded linear for any finite level of differentiability <cit.>. In particular, the space of functions with bounded Whitney derivatives up to order k form a Banach space, like for ordinary differentiability.

Let us now show that our function f are indeed Whitney differentiable on X_∞. The proof is elementary but is worthwhile understanding. We will keep the notation f_n for the restriction of f_n to X_∞.  

We first create a second sequence X'_n by starting on a slightly bigger disc D_ρ+, ρ+<1, and by removing from D_η disks D'_k ⊂ D_k of radius smaller than r_k, say for instance r_k/2, for k ≤ n,  so that
X_n ⊂ X'_n. 
The Cauchy inequalities give an estimate for the derivatives of a holomorphic function on X'_n restricted to X_n:

    sup_(x,y) ∈ X_n|_̣x^i_̣y^j g(x,y) | ≤2^j i! j!/r_k^j^isup_(x,y) ∈ X'_n| g(x,y) |


Using the estimate on the complement of  D'_k:
 
    |x^k/1+ky|<2 k^-1|x^k|<2 k^-1(ρ+)^k,

and the Cauchy estimates, one easily deduces  that,  for any k ≥ 0, the sequence (f_n) is a Cauchy sequence in the C^k-Whitney holomorphic sense  and therefore the sum function f is Whitney C^∞
in the holomorphic sense.

 By the Whitney extension theorem, the function f is the restriction to X_∞ of a standard C^∞ function

    F:^2 →

The set F=0 serves as C^∞-version of the limit correspondence defined by f=0.
As _̣y F(0)=_̣y f(0)=1 ≠ 0, the usual C^∞ implicit function
theorem shows that in a neighbourhood of the origin the set  F=0 is the
graph of a C^∞ function B, that we call a C^∞
Birkhoff-function attached to f(x,y). Clearly, the Taylor-series of B is b.

Somehow we proved more than expected, as the function B is now also defined for y<0. This may look surprising, if we think of the way the real parts of the curves V_n have multiple components which stretch along the horizontal lines
near the x-axis. The resolution of this paradox is that by removing the tubes
D_ρ× D_k, we removed the bendings of the curve at the same time.



 §.§ Content of the paper
 
If one wants to apply the Kleinbock-Margulis theorem, one should be able to define and establish the KMS non-degeneracy condition for the correspondence before going to the Birkhoff series and doing the Whitney extension. The purpose of this paper is to show that this is indeed the case.

That the Birkhoff series b does not see any of the bendings may explain
heuristically why establishing the KMS non-degeneracy condition based on considerations of b alone turns out to be very difficult. Our approach consists of working as much as possible with
the correspondences lying on the product space, rather than with
the implicit functions they define. So most of the paper concerns the definition and analysis of non-degeneracy conditions for correspondences.

In the last part, we explain the application of the results of this paper to
the Herman conjecture. We also hope that our result, elementary as it may look, is of interest independently of the context in which it originally emerged,
and that the deep connection between number theory and dynamical systems can
be further strengthened, in the spirit of Siegel, Arnold and Margulis. 
 
 
 

§ STATEMENT OF THE THEOREM

 

 §.§ Birkhoff functions, series and polynomials
  
 
  We consider the space ^N=^d ×^e with coordinates x=(x_1,…,x_d), y=(y_1,…,y_e) and projections:
  
 
    ^N [rd]^q [ld]_p    
    ^d       ^e

We then consider a decreasing chain X=(X_n) of locally closed subsets of ^N:

    X_0 ⊃ X_1 ⊃ X_2 ⊃…⊃ X_∞ :=∩_n X_n .

and assume that the sets X_n are equal to the closure of their interior:
X_n^∘=X_n.
 We also assume that X_n is of the form
 
    X_n=×_n

 where ⊂^d is a neighbourhood of the origin.

Such sequences arise often in constructions where
X_n+1 is obtained from X_n by removing some open subset.
Note that the closed set X_∞ in general could have empty interior.
We also assume to be given a formal series (in multi-index notations):

    f(x,y)=∑_I ∈^d  a_I(y)x^I, a_I ∈{ y }

where { y } denotes the ring of convergent power series in y=(y_1,…,y_e).
We identify the coefficients of the series with functions of z=(x,y) depending only on y. We use the multi-index notation and consider the Whitney C^-norms:

    a ^_U= sup_ z ∈ U ∩ X_∞ |^̣_y a(z)| , ∈^e

where the partial derivatives are taken only in the y-direction as the functions involved as coefficients of the series only depend on y.
  We say the series ∑_I ∈^d  a_I(y)x^I is X-convergent if
  there is a neighbourhood U of the origin
  such that for any ∈^e, the power series
  
    ∑_I ∈^d  a_I^_U x^I ∈[[x]]

 is convergent in p(U).

An  X-convergent series in U ∩ X_∞ defines a Whitney C^∞ function that we denote by the same name:

    f:U ∩ X_∞→, (x,y) ↦∑_I ∈^d  a_I(y)x^I

Note that for (x_0,y_0) ∈ U ∩ X_∞, the map-germ

    (^d,x_0) →, x ↦ f(x,y_0)

is an ordinary holomorphic map-germ.

 
We use the notation:

    V(f):={(x,y) ∈ U ∩ X_∞ |  f(x,y)=0}

where f=(f_1,…,f_k) are X-convergent series
and we let (V(f),0) be the germ of this set at the origin. 

A Birkhoff function-germ associated to  X-convergent series f=(f_1,…,f_e), where _̣y f(0) has maximal rank, is a C^∞-function germ: 

    B:(^d,0) → (^e,0)
 whose graph (denoted _B) satisfies: 

    (_B ∩ X_∞,0)= (V(f),0).


Birkhoff polynomials and Birkhoff series are defined in a similar way by considering infinitesimal  and formal neighbourhoods instead of germs at the origin.

Note that the Taylor expansion of a Birkhoff function B at the origin is the Birkhoff series of f. Indeed by the Whitney-Fefferman theorem (<cit.>), we may choose for any k>0 and any neighbourhood U of the origin a (real) continuous linear extension operator[We consider independently the real and imaginary parts when doing the extension.] :

    T:C^k(U ∩ X_∞,) → C^k(U,)

The series

    F(x,y)=∑_I ∈^d  T(a_I(y))x^I

defines a C^∞ function-germ.
 We have
  
    F(x,b_k(x))=o( x ^k)
   
  where b_k is the degree k Birkhoff polynomial of f.   The manifolds { y=B(x) }, { F=0 } and {y=b_k(x) } are therefore tangent at order k at the origin. Hence the degree k Taylor polynomial of the function B is the degree k Birkhoff polynomial.
  
 

 §.§ Non-degeneracy theorem
  
   
 If f=∑_I ∈^d  a_I(y)x^I is an X-convergent series then we may, by Whitney's theorem, extend f to a C^∞ function-germ F. By the implicit function theorem, the zero locus of F is then the graph of a Birkhoff function-germ. So Birkhoff functions always exists. The aim of this paper is to show that the Birkhoff function can be chosen to be KMS non-degenerate: 

 
 Assume that the formal series:

    f(x,y)=∑_I ∈^d  a_I(y)x^I

is X-convergent and that the matrix _̣y f(0,0):=(_̣y_j f_i(0,0)) is of maximal rank. Then
the series f admits KMS non-degenerate Birkhoff function germs, i.e., there exists Whitney C^∞ germs:

    B:(^d,0) → (^e,0)  with  (_B ∩ X_∞,0)= (V(f),0).




It is useful to work over the field of complex numbers but of course our constructions commute with complex conjugation, in particular if the a_I are real analytic and X invariant under complex conjugation then the function B can be chosen to be real, i.e.:

    B(x̅,y̅)=B(x,y)

Our construction is going to be explicit and not just an existence theorem.
 





§ MAPS, NON-DEGENERACY AND THE KLEINBOCK LEMMA





 §.§ The span

For a subset X ⊂^e, we denote by [ X ] the span of X, defined as the smallest affine subspace  containing X . 


Clearly, if X ⊂ Y, then [ X ] ⊂ [ Y ]. 
For a germ (X,p) of a set we define its span as

    [ X,p ]:=⋂_U ∈(p) [ U ∩ X ] ,

where U runs over the set (p) of open neighbourhoods of the point p.
We always can find a sufficiently small representative U for the germ (X,p),
so that  [ X,p ]=[ U ].


For an open set U ⊂^d and map g:U ^e we also define its
span as [ g(U) ], and we similarly define the span of a map germ
g:(^d,) → (^e,)̱ at  as the affine space

    [ g, ] :=∩_U [ g(U),]̱ ,

where the intersection runs over all representatives of the germ g.


 Let g: U ^e be an analytic map and U() the
  connected component of U containing . Then g(U(α)) is
  contained in the span of the germ of g at :

    g(U()) ⊂ [ g,  ].

Consequently,

    [ g(U()) ]=[ g,  ] .






 §.§ The jet-span of a map

Given a multi-index I ∈^d, we denote by

    ^̣Ig():=∂^I g/∂ x^I() ∈^e

the vector of I-th derivative of g at . We will also use the notation

    | I|=i_1+…+i_d, I=(i_1,…,i_d).


We define the m-jet span of a Whitney C^∞ map 

    g:X →^e

at a point ∈ X as the affine space:

    [ g,] _m:=[{^̣I g(): I ∈^d, |I| ≤ m} ]


The m-jet spans  form a increasing chain, so become
stable, at some minimal index τ=τ(g,):

    [ g, ]_τ=[ g,  ]_τ+1=…

We call  τ(g,) the torsion index of the map g at the point . 
The condition [ g, ]_τ=[ g, ] corresponds to the KMS non-degeneracy condition.  

As an illustration, consider the germ

    g:(,0) → (^4,0):x ↦ (1+x,x^3,x^100,x+x^3+x^100)=(y_1,…,y_4)

We have

    [g,0]_0 ⊂ [ g,0]_1 =[ g,0]_2 ⊂ [ g,0]_3= [ g,0]_4= … =[ g,0]_99⊂ [ g,0]_100=[ g,0]_101=….

The span [ g,0 ] is the affine hyperplane [ g,0]_100={ y_4=y_1+y_2+y_3+1 } and the map-germ has torsion index τ(g,0)=100 at the origin. It is KMS non-degenerate, as any holomorphic map.


A flat C^∞ map-germ like

    g:(,0) → (,0), x ↦{ e^-1/x^2 if Re  x> 0 
    
    0  otherwise.

has torsion index equal to zero at the origin but it is KMS degenerate. Indeed,
  the span of this map germ is [ g,0] =, but [ g,0]_n={ 0 } for any n therefore:
  
    [g,0]_n ≠ [ g,0 ]

   



 §.§ A lemma of Kleinbock


The following observation due to Kleinbock was formulated originally in
the real analytic setting:

  If g:(^d,)  (^e,)̱ is a holomorphic map germ, then it  is KMS non degenerate.


For simplicity assume that =̱0 so that the span is a vector space.
Let g:U ^e be a connected representative of the germ (g,)
The sequence of spans [ g, ]_n generated by the vectors ^̣Ig(), |I| ≤ n stabilises at level τ. Clearly [ g, ]_τ⊂ [ g(U) ]. To show the other inclusion, let ℓ be an affine linear form vanishing on [ g, ]_τ. Then the Taylor series of ℓ∘ g at
 vanishes identically and therefore the holomorphic function ℓ∘ g vanishes on all of U. As this holds for all ℓ defining [ g,  ]_τ, we find [ g(U)] ⊂ [ g,  ]_τ, hence we have equality.


As we saw previously, in the C^∞ case, we may very well have a proper inclusion:

    ⋃_k [ g, ]_k ⊂ [g,]  but ⋃_k [ g, ]_k ≠ [g,].







§ CORRESPONDENCES AND NON-DEGENERACY




 §.§ Maps and correspondences

Many naturally appearing expressions like x^2+y^2=1 define y only
as a multivalued function y=√(1-x^2). Especially in the complex
domain, insisting on working with a monovalent function is cumbersome and
not very enlightening. Therefore, in algebraic geometry one is used to
working with correspondences, which provide a convenient extension of
the usual strict concept of function. The Poincaré example shows that this point of view is suited to the type of objects we are considering.  So our purpose will be to adapt the previous notions defined for maps to correspondences.







In this paper we will use the following terminology:


  A complex analytic variety V ⊂^N=^d ×^e is said to
  define a regular correspondence at ø=(,)̱, if the following
  conditions
  are verified:

  1) the set V is smooth of dimension d at ø.

  2) Under the projection onto the first factor, the tangent space T_øV
 maps isomorphically onto ^d.


Of course, by the implicit function theorem, we then can find a local
parametrisation g:(^d,)  (^e,)̱, x ↦ y=g(x)
so that (x, g(x)) ∈ V, or (Γ_g,ø) =(V,ø), where Γ_g
denotes the graph of g.




 §.§ The relative span

For a subset X ⊂^N:=^d ×^e, we denote by X the relative span of X, defined as the smallest affine subspace of the form ^d × F containing X where F ⊂^e is an affine subspace.
Clearly, one has

    X = q^-1(q([X])) ,

where q:^d ×^e ^e is the projection on the second factor.

Of course, the notions of span of a set and of a map germ are related:

Let Γ_g ⊂^d ×^e=^N denote the graph of
g:U ^e and let 
q:^N →^e be the projection onto the second factor. Then


    [ g(U)] =q(Γ_g)
  


 Let ℓ:  ^e → be an affine function vanishing on
[ g(U)] then

    ℓ':  ^d ×^e →, (x,y) ↦ℓ(y)

is an affine function vanishing on _g and conversely.
 


We define the relative span at a point ø∈ X
as the affine space

    X,ø  =∩_U  X ∩ U,ø ,

where the intersection runs over all neighbourhoods of ø∈^N and get an analog of Lemma <ref>:

 Let V ⊂^N be an analytic set and V(ø) the
connected component of V that contains the point ø∈ V. Then V(ø)
is contained in the span of the germ (V,ø):

    V(ø) ⊂ V, ø

and consequently:

    V(ø)= V, ø .




Let ℓ be an affine function vanishing on V, ø. As a holomorphic
function vanishing on the germ (V,ø), it vanishes on a neighbourhood of V
containing ø. By the identity theorem of analytic functions, ℓ vanishes
on the connected component V(ø).




 §.§ The relative jet-span of an analytic set

For a germ of an analytic set (V, ø) we define the n-jet span 

    V, ø_n

 as the smallest affine subspace of the form ^d × F that contains the n-th infinitesimal
neighbourhood of the fibre at ø∈ V of the projection ^N →^d. It is alternatively defined by the kernel of the affine linear functions contained in the ideal I+M_ø,x^n+1 which depend only on the y-variables. Here y_1,…,y_e denote coordinates on ^e, I ⊂_^N,p
is the ideal defining (V,ø) and M_ø,x⊂_^N,ø the maximal
ideal of holomorphic functions vanishing at x=, ø=(,)̱. 

Clearly, one has


    V, ø_1  ⊂ V, ø_2 ⊂ V, ø_3 ⊂…,

so this chain stabilises for a certain index τ:=τ(V,p), that we call the  torsion index of V at p:

    V, ø_τ  =  V, ø_τ+1=…

 
 For any holomorphic map-germ g:(^d,) → (^e,)̱ we have

    q(_g,ω_m)=[g,]_m

for ø=(,)̱ and any m ∈.


Write _g,_m={ℓ=0 } where 

    ℓ:^d ×^e →^r, (x,y) ↦ℓ(y)
 is some affine map depending only on the y-variables. By the Taylor formula, we have

    ℓ(g(x))=∑_ |I| ≤ m^̣Ig()/|I|!(x-)^I+o( x ^m)

and therefore the conditions ℓ(g(x))=o( x ^m) and ^̣Ig()=0 for |I| ≤ m are equivalent.














 §.§ A variant of the Kleinbock lemma



The following result is the analog of the Kleinbock lemma for analytic spaces:

 
 Let (V,ø) ⊂ (^N,0) be a germ of an analytic space with torsion index τ at ø then:

    V,ø =  V,ø_τ




  Clearly, V,ø_τ⊂ V,ø. For the converse, note that we have
  ∩_n M_p^n =(0), so that a function that vanishes on all infinitesimal
  neighbourhoods of ø vanishes on (V,ø), from which we obtain the reverse inclusion.
 







 §.§ n-Correspondences and the finite determinacy lemma

We now consider truncated sums as we did in the introduction for the Poincaré example:
  A correspondence V is called an n-correspondence at ω,
if it is defined by d equations of the form   

    f(x,y) =∑_|I| ≤ n, I ∈^d a_I(y)x^I ∈{ y}[x]

 
 


 



 
 The following lemma is a refinement of Lemma <ref>:

 
  
  If V is an n-correspondence at ω, then 
  
    τ(V,ω) ≤ n

  that is: V,ω_n = V, ω.


  
Consider an affine function ℓ vanishing on
  V,ω_n  depending only on the y-variables.
 Let v ∈^d be an arbitrary vector and consider the function
  
    :(,0) → (^e,0), t ↦ f(+tv,y)

  By definition of the n-span,  the polynomial ℓ∘ in the variable t has a zero of multiplicity ≥ n+1 at the origin:
  
    (ℓ∘)^(k)(0)=0, k=0,1,2, …,n.

  
  As (V,ø) is an n-correspondence, the equation ℓ∘(t)=0 has degree ≤ n therefore
    ℓ∘ vanishes identically. As the vector v is arbitrary, we deduce that ℓ vanishes on (V,ω). 
  This inclusion being true for any hyperplane containing V,ω_n,
  we  get that V,ø⊂ V,ω_n.
  This proves the lemma.
  
 
  


 §.§ The spans stabilise

We return to the situation of  subsection <ref> and consider a decreasing chains of subsets of ^N=^d ×^e:

    X_0 ⊃ X_1 ⊃ X_2 ⊃…⊃ X_∞ :=∩_n X_n .

with
X_n^∘=X_n  
 and consider a formal power series
 
    f(x,y) =∑_ I ∈^d a_I(y)x^I ∈{ y} [[x]], f(0)=0

 and define
 
    f_n(x,y) =∑_|I| ≤ n, I ∈^d a_I(y)x^I ∈{y}[x]

We denote their
zero-sets by

    V_n:={(x,y) ∈ X_n |  f_n(x,y)=0}⊂ X_n


We also have to consider

    V_∞:={(x,y) ∈ X_∞ |  f(x,y)=0}⊂ X_∞







One has  
    V_n,0⊂ V_n+1,0 .


 We have
  
    V_n,0 = V_n,0_n= V_n+1,0_n⊂ V_n+1,0_n+1= V_n+1,0.

 For the first and last equalities we used that V_n is an n-correspondence
  (for n and n+1), the second equality is due to the fact that
  
    f_n+1(x,y)=f_n(x,y)+O(x^n+1)
 The middle inclusion is
  by the definition of jet-span.


As a consequence, the sequence V_n,0 stabilises, say at level ρ
and we define

    W_∞ := V_n,0, n ≥ρ .



For n ≥ρ, we have:

    V_∞,0⊂ V_n,0= W_∞


  

    Take (x_0,y_0) ∉ W_∞. By definition of W_∞, this means that for any n ≥ρ,
    the point (x_0,y_0) does not belong to V_n,0 and therefore is not in V_n, so we have
    f_n(x_0,y_0) ≠ 0 for n ≥ρ. Consequently, for n ≥ρ, the holomorphic map-germs
   
    ϕ_n:(^d,x_0) →^e, x ↦ f_n(x,y_0)=∑_|I| ≤ n a_I(y_0)x^I

are  non-zero at x=x_0.

 We now use the classical Hurwitz lemma (see e.g. <cit.>): Consider a sequence of holomorphic functions ϕ_n converging in the C^0-topology and assume that the limit ϕ_∞ is not the zero-function. Then:
  if the functions ϕ_n are non-zero at a point, then the limit ϕ_∞ is also non-zero at this point
  [This Hurwitz lemma, sometimes called the Hurwitz theorem, is a straightforward consequence of the Cauchy integral formula.].

As f_n(x_0,y_0) ≠ 0, the coefficients a_I are not all equal to zero and therefore the holomorphic map-germ

    ϕ_∞:(^d,x_0) →^e, x ↦∑_I ∈^d a_I(y_0)x^I

is not identically zero. From the Hurwitz lemma we conclude that  (x_0,y_0) ∉   V_∞ and therefore  
    (V_∞,z_0) ⊂ W_∞, z_0 =(x_0,y_0)
 


 

 §.§ Proof of Theorem <ref>

 We keep the same notations and let 
 
    g:(× W_∞,0) → (^e,0)

 be the restriction of f to W_∞. By Lemma <ref>,
 we have 
 
    (V(f),0)=(V(g),0).
 We choose a Whitney extension G of g. By the implicit function theorem the zero locus of G is the graph of a function-germ 
 
    :̱(^d,0) → (W_∞,0), x ↦(̱x)

 and we may extend the image to ^e to get a Birkhoff function

    B:(^d,0) → (^e,0), x ↦(̱x)

  The Taylor expansion of B at the origin is the Birkhoff series whose partial derivatives generate W_∞
  by definition. Hence the function B is KMS non-degenerate. This concludes the proof of the theorem.



§ APPLICATION TO THE HERMAN CONJECTURE

We give a brief outlook of our proof of the Herman invariant tori conjecture, details will be published elsewhere, the reader may consult <cit.> for further details.


 §.§ The frequency map

In KAM theory, the invariant tori of a Hamiltonian system are parametrised by the frequencies of motions which are not very well approximated by the standard integral lattice ^e.
In order to measure how a frequency vector y ∈^e is approximated by the lattice, one considers the collection of resonant hyperplanes (_) with 

    _={ y ∈^e: (y,) =0 }, ∈^e

where 

    (y,)=∑_j=1^e y_j _j

is the complexification of the Euclidean scalar product. We then fix a real positive sequence
(_), ∈^e, which converges to zero not too fast
and consider tubular neighbourhoods of hyperplanes

    T_={ y ∈^e: |(y,)| < _}, ∈^e

 We now define the sequence (X_n) by:

    X_n=^e ∖⋃_≤ n T_

and this set parametrises the invariant tori, in the sense that there is a Birkhoff function

    B: U →^e

called the frequency map whose fibres above points of X_∞=⋂_n X_n are (complexifications) of invariant tori. In <cit.>, we constructed B as a map obtained from a
correspondence as we now explain.



 §.§ The small denominators ring

We consider the polynomial ring [ω]=[ω_1,ω_2,…,ω_d] and construct a ring SD_α of small denominators at α∈^d. The ring SD_α is a subring of the field (ω)=(ω_1,ω_2,…,ω_d) of rational functions 
 defined by 
localisation of [ω]=[ω_1,ω_2,…,ω_d] with respect with the multiplicative subset S generated by all linear polynomials (+ω, J), J ∈^n∖{0}:

    SD_α:=[ω]_S:=[ω,1/(+ω,J), J ∈^n∖{0}]
    ⊂(ω) .

The elements of this ring may have poles along the resonance hyperplanes

    H_J:={ω∈^d: (+ω,J)=0 }

We will construct a normal form inside the algebra

    R:=SD_α[[τ,p,q]] ⊂[[ω,τ, p,q]],

a subalgebra of the power series in 4d variables 

    ω_1,ω_2,…,ω_d,τ_1,…,τ_d,q_1,…,q_d,p_1,…,p_d .

We provide R with the standard Poisson bracket, so that the Poisson centre of
R is the ring

    R_0:=SD_[[τ]] ⊂[[ω_1,…,ω_d,τ_1,…,τ_d]].

We endow this ring of quasi-homogenous filtration with weights

    w(q_i)=w(p_i)=1, w(τ_i)=2, w(ω_i) =0

We use the notations [f]_i^j for the sum of terms of weighted degree ≥ i and <j and omit the indices if i=0 or j=+∞.
We also use the notation O(n) for terms of degree ≥ n.



 §.§ The Hamiltonian normal form iteration

Starting from a Hamiltonian 

    H=∑_i=0^d α_i p_iq_i+O(3),

we first form the ω-extension of H:

    F_0=∑_i=0^d (α_i+ω_i) p_iq_i+O(3),


Then we solve a homological equation of the form:

    v_0(A_0)=[F_0]_3^4 +t_0,    t_0 ∈  R_0+ I^2 ,
 
where I is the ideal generated by the  p_iq_i-τ_i and v_0 is a Poisson derivation.
The definition might seem involved but it produces a sequence of biholomorphic maps e^-v_i which are defined globally and not "blocked by resonances."

Exponentiating the Poisson derivation v_0, we produce a Poisson automorphism  e^-v_0. The application of e^-v_0 transforms F_0 into F_1, where the cubic term is  removed modulo an element of R_0+I^2 ; we put A_1=A_0+t_0. For the n=0 case, it turns out that t=0 but at the next level we have to solve

    v_1(A_1)=[F_1]_4^6 + t_1 ,    t_1 ∈ R_0+I^2
 
for the degree 4 and 5 part of F_1 on A_1 and, as a general rule t_1 ≠ 0. 

Then the application of e^-v_1 to F_1 produces F_2, where now these terms of degree 4 and 5 are removed, but certain terms in R_0+I^2 are introduced. These remaining terms we add to A_1 and obtain A_2. Next we solve the homological equation for the terms of degree 6,7,8,9 of F_2, but now on  A_2, etc. Thus we obtain a by iteration a sequence of triples

    (F_n, A_n, v_n ),    n=0,1,2,…

We have correspondences defined by the rational functions 

    [e^-v_n… e^-v_0ω_i]^2^n= ∑_, || ≤ 2^n a_(ω) τ^, a_∈ SD_

 
 precisely of the kind envisioned by Poincaré!  This power series is X-convergent, the partial sums define complex variety and the limit variety
 parametrises the pairs consisting of invariant tori and associated frequencies. 
 
 The theorem of this paper shows that the limiting correspondence contains the graph of a KMS
 non-degenerate map.  Using an adequate variant of the Kleinbock-Margulis theorem (<cit.>), one deduces the fact that the invariant tori form a set of positive measure.  The convergence
 of the iteration scheme proposed  is subtle, but it turns out that it can be
 streamlined using the theory of Banach functors developed in <cit.>.  In that
 set-up, the non-degeneracy argument is easy and, in fact, as this paper shows, “almost trivial”. 

 
amsplain



 Paper Title: Global stability in the Ricker model with delay and stocking.






Global stability in the Ricker model with delay and stocking
    Ziyad AlSharawiCorresponding author: zsharawi@aus.edu  and  Sadok Kallel 

Department of Mathematics and Statistics

 American University of Sharjah, P. O. Box 26666

  University City, Sharjah, UAE
 
    August 5, 2024
============================================================================================================================================================================================================



§ ABSTRACT 
We consider the Ricker model with delay and constant or periodic stocking.  We found that the high stocking density tends to neutralize the delay effect on stability. Conditions are established on the parameters to ensure the global stability of the equilibrium solution in the case of constant stocking, as well as the global stability of the 2-periodic solution in the case of 2-periodic stocking.  Our approach extensively relies on the utilization of the embedding technique. Whether constant stocking or periodic stocking, the model has the potential to undergo a Neimark-Sacker bifurcation in both cases. However, the Neimark-Sacker bifurcation in the 2-periodic case results in the emergence of two invariant curves that collectively function as a single attractor. Finally, we pose open questions in the form of conjectures about global stability for certain choices of the parameters.

AMS Subject Classification: 39A10, 39A30, 92D25

Keywords: Ricker model, global stability, embedding, periodic solutions, stocking.


§  INTRODUCTION

As an early attempt to model a population of a single species with non-overlapping generations, Moran (1950) <cit.> proposed using a one-hump map that increases to a maximum and then decreases asymptotically to a non-negative value. Along this line of thought and based on experimental data, Ricker introduced a density-dependent model that became known in the scientific literature as the Ricker model <cit.>

    x_n+1=x_nf(x_n)=x_ne^r(1-x_n/K),    K, r>0  and x_0∈_+=[0,+∞).

In this model, the parameters K and r represent the carrying capacity and the intrinsic growth rate, respectively. To learn more about the origin and significance of Ricker's model, and Ricker's pivotal contribution in the field of quantitative fishery science, we refer the reader to <cit.>.  Rescaling can reduce the model to y_n+1=y_nexp(r-y_n).  This model enjoyed a boost of success after the publication of the two studies by May <cit.>, and a third study by May and Oster <cit.>, in which the model's intriguing dynamics were investigated, and the global stability of its positive equilibrium for 0<r<2 was observed. A rigorous proof of the global stability for 0<r≤ 2 was developed later using various techniques such as Lyapunov functions <cit.>, Singer's theorem <cit.> and enveloping <cit.>.



While implicit time lags are embedded in discrete systems, explicit time lags caused by significant recruitment delays must be accounted for in the density-dependent function f. Time delays are necessary to accommodate the recruitment to the adult stage, which may vary based on the type of species  <cit.>. This mechanism requires that a mathematical model be considered with a certain time lag N. As a result, Eq. (<ref>) becomes

    y_n+1=y_nf(y_n-N)=y_ne^r-y_n-N.

Due to implicit delays, solutions of discrete models are observed to overshoot and undershoot their equilibrium level and generate oscillations, periodic solutions, or chaos. This tendency is typically aggravated when explicit delays are incorporated into the system <cit.>. When a population exhibits convergence to its equilibrium regardless of its initial density, the phenomenon is called global stability. Proving mathematically that the equilibrium solution is globally stable becomes challenging when time delays are involved. Under general settings of the density-dependent map f, Liz et al. <cit.> found sufficient conditions to obtain global stability for Eq. (<ref>) when 0<r<3/2(N+1). This implies 0<r<3/4 when N=1.  In a lengthy and technical paper that included computer-assisted proofs, Bartha et al. <cit.> established global stability when N=1 and 0<r<1.



Stocking or harvesting can be used as a management tool to achieve various goals, including chaos reversal and re-stabilization of an equilibrium level <cit.>. According to <cit.>, constant rate harvesting is observed to transform contest competition into scramble competition. On the other hand, constant stocking slows the fluctuation and has a stabilizing effect on equilibrium solutions <cit.>.  When constant stocking is applied to the Ricker model with delay,  Eq. (<ref>) becomes

    y_n+1=y_nf(y_n-1)+h=y_ne^r-y_n-1+h=F(y_n,y_n-1),   h>0.

The parameter h can also be regarded as a perturbation parameter, as investigated in  <cit.>, but without considering any time delay. Our primary motivation for considering Eq. (<ref>) stems from two significant factors, namely the effect of delay and stocking on stability. The individual impact of delay and stocking was considered in research <cit.>, and it is known to reflect contrasting impacts on stability. However, exploring the interaction between delay and stocking and their impact on overall stability is a novel task. 



Equations of the form y_n+1=y_nf(y_n-1)+h have been considered in <cit.> under the assumption that tf(t) is increasing; however, Eq. (<ref>) does not belong to this category.  If the stocking is done through seasonal quotas, the constant h is replaced by a sequence {h_n}, which we assume to be p-periodic.
 Therefore, Eq. (<ref>) becomes

    y_n+1=y_nf(y_n-1)+h_n=y_ne^r-y_n-1+h_n=F_n(y_n,y_n-1),

where h_n≥ 0 is a p-periodic sequence and F is defined on the positive orthant ^2_+. We continue to write f(y) instead of e^r-y whenever we find it convenient. Unlike Eq. (<ref>), which has two equilibrium points x=0 and x=r, Eq. (<ref>) has a unique non-negative equilibrium solution, which bifurcates into a p-periodic solution when periodicity is introduced in Eq. (<ref>).



The structure of our paper is focused on the development of the theory and apparatus required to investigate the global stability of the equilibrium solution in Eq. (<ref>) and the p-periodic solution in   Eq. (<ref>). We provide a clear geometric understanding of the reason why stability (local or global) holds. The sections of this paper are structured as follows: In Section Two, preliminary findings regarding the utilization of the embedding technique in autonomous and periodic cases are presented.
 In the third section, we leverage the outcomes from Section Two to apply them to Equation (<ref>). This section delineates sufficient conditions on the variable h that ensure global stability in the autonomous case. We determine “almost completely" the local and global stability regions in the (h,r) plane, modulo a well-identified restricted region of uncertainty. In Section Four, we turn our attention to Equation (<ref>) and employ the embedding technique to investigate the 2-periodic case, ultimately establishing the global stability of the 2-periodic solution under certain sufficient conditions.
 This paper closes with a conclusion section that not only summarizes our key findings but also raises pertinent open questions for further exploration.




§ THE EMBEDDING TECHNIQUE 

Embedding a dynamical system in general into a higher dimensional dynamical system that can be utilized to classify certain characteristics of the original system is a known approach <cit.>. However, the challenge arises when it comes to the technical details of the approach, particularly with the partial order in the embedded system.  In this section, we build the basic machinery needed in the sequel.  Let V denote a partially ordered metric space, specifically the positive orthant, as defined for the purposes of this discussion, _+^n or [a,b]^n for some n.
A recursive sequence in V, with delay k, is any sequence defined by

    α_F : x_n+1 = F(x_n,…, x_n-k+1),   k≥ -1,

where F:V^k→ V is a continuous function and the initial terms x_0,x_-1,…, x_-k+1 are given in V. These initial terms, together with F, determine the sequence uniquely.
We write 𝒮(V), the set of all recursive sequences in V. This can be topologized as a subspace of
V^w, the infinite direct product of V. Notice that different functions F can give rise to the same recursive sequence (<ref>), so only the sequence uniquely determines the system.

 A continuous injection Ψ: 𝒮(V_1)↪𝒮 (V_2), which sends convergent
sequences to convergent sequences, is called an “embedding”. If V_1 is a subspace of V_2, then
there is a canonical inclusion 𝒮 (V_1)⊂𝒮 (V_2).

It is pertinent to note that, generally, the convergence of Ψ (α) does not entail the convergence of α for a given embedding Ψ: 𝒮(V_1)→𝒮(V_2).




 §.§ Embedding in the autonomous case

Consider the two-dimensional difference equation
α_F: x_n+1=F(x_n,x_n-1), where F:^2_+→ is any continuous function.  We define T(x,y) = (F(x,y),x), so that
T(x_n,x_n-1)=(x_n+1,x_n) and its iterations produce a sequence in the plane

    (X_n) : (x_0,x_1), (x_1,x_0),…, X_n:= (x_n,x_n-1),

which entirely describes
the dynamics of the system.  We refer to  T as the “vector form” of the system α_F.
The system T has a global attractor, which means that the sequence (X_n) converges independently of the choice of the initial value within the given domain.
The assignment  (x_n)↦ (X_n)
gives an embedding Ψ: 𝒮()↪𝒮(^2). This is a strong embedding in the sense that Ψ(α) is convergent if and only if
α is convergent.



Next, we write V=^2_+. For every sequence α_F∈𝒮(V ),  we define a 4-dimensional sequence ζ_F∈𝒮(V^4) with general
term ζ_n = (X_n,X_n)= (x_n,x_n-1,x_n,x_n-1), and initial term ζ_0 = (x_0,x_-1,x_0,x_-1). This is the image of (X_n) under the diagonal embedding
V^2→ V^2× V^2, (x,y)↦ (x,y,x,y).
This is a strong embedding as well 𝒮(V)↪𝒮(V^4), which we refer to as the “diagonal embedding”. Because this is a strong embedding, a good way to establish convergence of ζ_F will help us obtain convergence in our initial system.

Let us write ζ_F in recursive form as follows. Define the self-map of V^4 as

    G(x,y,u,v) = (F(x,y),u, F(u,v), x).

Starting with ζ_0, we see that
ζ_n = G^n(ζ_0).
The advantage of introducing the map G is to obtain a form of monotonicity <cit.>.
Introduce the “southeast partial ordering" on V× V by (x_1,y_1)≤_se (x_2,y_2) if and only if x_1≤ x_2 and y_1≥ y_2.  Using the southeast partial ordering on V^2× V^2 means using ≤_se on W× W with W=V^2.
A key observation now is that if F is non-decreasing in its first component and non-increasing in its second one, i.e., F(↑, ↓), then the sequence G is monotonic with respect to the ≤_se partial ordering on V^2× V^2. More precisely, rewrite  G:V^4→ V^4 as a map
G: V^2× V^2→ V^2× V^2, (X,U)⟼ G(X,U). Then

    (X_1,U_1)≤_se (X_2,U_2) ⟹
    G(X_1,U_1)≤_seG(X_2,U_2).

This monotonicity gives a handy way of studying the convergence of the diagonal sequence ζ_F: ζ_n=G^n(ζ_0), and thus ultimately that of α_F.  We develop this approach next.

 Write A=(a,b) and B=(b,a). We say that X=(x,y) is inside a box with vertices A and B in V^2 if a≤ b and A≤_se X≤_se B. Equivalently, if in V^4 we have

    (A,B)≤_se (X,X)≤_se(B,A)



Suppose that for a choice of the map F and the points A and B, the map G in Eq. (<ref>) has the property that
(A,B)≤_seG(A,B) and G(B,A)≤_se (B,A). This happens if (a,b)≤_se (F(a,b),F(b,a)). Then, if X_0 (resp. some X_k=T^k(X_0) for some k≥ 0)
is inside a box with vertices A,B,  we can see immediately that for positive n,

    (A,B)≤_se G^n(A,B)≤_se G^n(X_0,X_0)≤_se G^n(B,A)≤_se(B,A),

which means that our sequence ζ_F is eventually caught between the orbits of G through (A,B) and (B,A). The sequences G^n(A,B) and G^n(B,A), being bounded and monotonic in the southeast ordering, converge to fixed points of G.
A fixed point of G must be of the form

    x̅=(x,y, y, x), where  x = F(x,y) and  y=F(y,x).

Such fixed points come in pairs if x≠ y, and so if  (F(x,y), F(y,x)) = (x,y) has a unique solution,
then x=y=x̅ is a fixed point of F. If x≠ y, then (x,y) and (y,x)  are dubbed as pseudo fixed points of F (these were called “artificial fixed points” in <cit.>). The above discussion leads us to the following result <cit.>.

 Let F: V^2→ V, where F(↑,↓) and consider the system x_n+1=F(x_n,x_n-1) with an initial condition X_0= (x_0,x_-1)∈ V^2. Suppose there is (a,b)∈ V^2 such that (a,b)≤_se(F(a,b),F(b,a)) and (a,b)≤_se (x_k,x_k-1)≤_se(b,a) for some k≥ 0. If (F(x,y), F(y,x)) = (x,y) has a unique solution in V^2, then
ζ_F converges to a global attractor in V^4 and α_F converges to a global attractor in V^2. If (F(x,y), F(y,x)) = (x,y) has three solutions (x^*,y^*), (y̅,y̅) and (y^*,x^*), x^*<y^*, then

    x^*≤liminfα_F≤limsupα_F≤ y^*.




A  convenient way of stating this result is as follows: Recall that
X_n=T^n(x_0,x_-1)= (x_n,x_n-1). Suppose F(↑, ↓) and (a,b)≤_se(F(a,b),F(b,a)). Suppose that the sequence
(X_n) is eventually in the box with vertices A=(a,b) and B=(b,a). If G: V^4→ V^4 has a unique fixed point, then it must be of the form (x,x,x,x) and α_F must converge to x. Alternatively, the system T has the global attractor (x,x) in that box.


A q-cycle C_q:={ξ_0,ξ_1,…,ξ_q-1} of the map G in Eq. (<ref>) is a periodic solution of the system ξ_n+1=G(ξ_n) with minimal period q.  Note that C_q could be driven by a q-cycle {x_0,x_1,…,x_q-1}  of F, where 
    ξ_0=(x_0,x_q-1,x_0,x_q-1)  and  ξ_j=(x_j,x_j-1,x_j,x_j-1),  j=1,…, q-1.
 Otherwise, we say that F has a pseudo q-cycle. This notion has been introduced and classified in <cit.>.




 §.§ Periodic systems


We consider in this section dynamical systems of the form

    α : x_n+1 = F_n(x_n,x_n-1) , (x_0,x_-1)∈ V^2,

where we assume F_n+p=F_n, ∀ n≥ 0, and p≥ 1 is the minimal positive integer with such a property. We refer to such a sequence as being “p-periodic”. These sequences are treated in <cit.>, and we give a general overview next.
We stress that the starting time n=n_0 is crucial in non-autonomous systems because it dictates certain synchronization between time and space. In this paper, we consider n_0=0 throughout.   The sequence α := {(x_n)}_n≥ 0 in V gives rise to the sequence (X_n):= {(x_n,x_n-1)}_n≥ 0 in V^2 as before. To stress the role of the individual maps in the system (<ref>), we denote it by [F_0,F_1,…,F_p-1] and its vector form by [T_0,T_1,…,T_p-1], where
T_n(x_n,x_n-1):= (F_n(x,y), x) , n≥ 0.
Considering first the period two case, let us write T_01=T_0∘ T_1 and T_10=T_1∘ T_0 the compositions whose associated sequences in V^2 are given by

    (X_2n+1): (x_2n+1,x_2n) = T_01(x_2n-1,x_2n-2)  and  (X_2n): (x_2n,x_2n-1) = T_10(x_2n-2,x_2n-3)
 with initial values  (x_0,x_-1) and (x_1,x_0) respectively.
These two sequences are interlocked as in Figure <ref>.



Decomposing the 2-periodic recursive sequence (X_n) into these two subsequences give an embedding
𝒮(V)→𝒮(V^2)×𝒮(V^2) whose image is
(X_2n+1,X_2n). Concatenating these sequences side-by-side produces a recursive sequence ζ = Ψ (α) in 𝒮(V^4) given by the general term

    ζ_n:= (x_2n+1, x_2n, x_2n,x_2n-1) where ζ_0 = (x_1,x_0, x_0,x_-1).


The above is not a strong embedding in general since a convergence of a pair of subsequences implies convergence of the original sequence unless they both converge to the same limit. In all cases, this construction extends in an obvious manner to p-periodic recursive sequences of finite delay and gives an embedding of the subset of recursive p-periodic sequences of delay N into 𝒮(V^N).

Starting with Eq. (<ref>), p=2, the system in vector form is written [T_0,T_1], and it breaks into the two “folded” systems T_10 and T_01 as already indicated.
We will relate the k-cycles of T_10 and T_01 to the 2k-cycles of [T_0,T_1]. Note that this process is known as folding and unfolding, and it has been characterized for one-dimensional maps <cit.>.


 Define α : x_n+1 = F_n(x_n,x_n-1), (x_0,x_-1)∈ V^2 which is 2-periodic.
 Let C_k(T_01)⊂ (V^2)^k (resp. C_k(T_10)) describe the set of k-cycles of T_01 (resp. T_10). Then T_0 maps C_k(T_01) bijectively onto C_k(T_10), with inverse T_1. If these sets are disjointed, their union is a 2k-cycle of [T_0,T_1].  In particular, [T_0,T_1] has a unique 2-cycle if and only if T_01 and T_10 have unique fixed points which are distinct.


   This is an immediate consequence of the identities T_1∘ T_01 = T_10∘ T_1, and T_0∘ T_10 = T_01∘ T_0.
The final claim is a consequence of the fact that the fixed points of α correspond to its one cycle.


 In the next two illustrative examples, we ignore monotonicity, and focus on the structure of cycles.

Let p=k=2.
Here's an example where the 2-cycles for T_01 and T_10 are different.
Define α as in Eq. (<ref>) with
F_0(x,y)= y+x^2-1 and F_1(x,y)= -y. Then

    T_0(x,y) = (y+x^2-1, x) and  T_1(x,y) = (-y,x).

We can check that {(1,y), (-1,y)} is the unique 2-cycle of T_10=T_1T_0, while {(y,1), (y,-1)} is the unique 2-cycle of T_01=T_0T_1. Clearly,
T_0(1,y)=(y,1) and T_1(y,1)=(-1,y). The system [T_0,T_1] has a 4-cycle.



Define α as in (<ref>) with
F_0(x,y) = xy and F_1(x,y)=x y. This is 2-periodic, with subsequences in vector form induced from T_0(x,y) = (xy,x) and T_1(x,y) = (x y,x). A  calculation shows that both systems  T_10 and T_01 have each a unique three cycle that is  common to both

    {(-1,-1), (1,-1), (-1,1)} =  C_3(T_10)=  C_3(T_01).

This is also the same 3-cycle for each individual map T_j and for the 2-periodic system [T_0,T_1]. This is due to the fact that the periodicity of the system 2 and the periodicity of the cycle 3 are coprime numbers.


Similarly, treating the p-periodic case for any p≥ 2 is possible. In that case, the sequence α in (<ref>) gives rise to p sequences in V^2, indexed by the action of the cyclic group ℤ_p with generator σ acting on
the ordered tuple (p-1,p-2,…, 1,0)  by cyclic permutations of coordinates. Here σ^i(p-1,p-2,…, 1,0) = (i-1,i-2…, 0,…, i).
To this corresponds the sequence
X_σ^i with general recursive term

    (x_n+1,x_n) = T_σ^i(x_n,x_n-1)  where  T_σ^i=T_i-1 T_i-2⋯ T_0⋯ T_i,

T_i(x,y)=(F_i(x,y), x), with initial term T_i-1⋯ T_0(x_0,x_-1).
The sequences X_σ^i partition (X_n). They are mirror copies of each other in the sense that either all converge simultaneously or diverge. If some sequence X_σ^i has a k-cycle, then all other sequences have k-cycles. If some sequence has a global attractor, so do all other sequences (see <cit.>, Lemma 2.1).
It is of potential interest to understand how k-cycles of the T_σ^is' produce q-cycles of [T_0,T_1,…, T_p-1]. The k-cycles in C_k(T_σ^i), for different i, may overlap or even be equal as illustrated in Example <ref> and Example <ref>. Note for example that if the F_i's are injective, then a fixed point of T_σ^i gives rise to a p-cycle of [T_0,T_1,…, T_p-1]. This notion is a straightforward generalization of the one-dimensional case in <cit.>.







 §.§ Embedding in the periodic case
 
In the autonomous case (Subsection <ref>),
we extended a one dimensional system α_F as given in Eq. (<ref>) into a
a four dimensional recursive system ζ_F : ζ_n+1 = G(ζ_n), where G is a self map of V^4. When working in a box (see Definition <ref>),  if G had a unique fixed point, then α_F converged to a unique fixed point. We seek an analogous result in the periodic case, i.e., Eq. (<ref>). It turns out in this case that a unique fixed point of some corresponding embedded system in V^4 leads to the existence of a globally attracting cycle of [T_0,T_1] (and α_F). We will restrict ourselves below to the 2-periodic case, i.e., p=2.

Starting with (<ref>), F_0 , F_1, F_1≠ F_0 and F_i(↑,↓), define similarly

    G_0(x,y,u,v) = (F_0(x,y),u, F_0(u,v), x)  ,  G_1(x,y,u,v) = (F_1(x,y),u, F_1(u,v), x).

We write G_01 = G_0∘ G_1 and G_10= G_1∘ G_0.  The four dimensional embedded system [G_0,G_1] is defined by  ζ_n+1 = G_n 2(ζ_n). Again, the restriction of [G_0,G_1] to the diagonal subset in V^2× V^2 is the “diagonal system”
([T_0,T_1],[T_0,T_1]). The following is a direct check.

 
Assume that F_i(↑,↓) are strictly monotonic in each component. Then both G_0 and G_1 are one-to-one.


A fixed point of the system [G_0,G_1] must be a fixed point of all the G_i's. In other words, if ζ̅∈ V^4 is a fixed point of [G_0,G_1], then

    ζ̅ = (,,,),  where 
    F_j(, ) =  and  F_j(,) = , j=0,1.

Note that a fixed point of
[G_0,G_1] is necessarily a fixed point of both G_01 and G_10. The converse is not always true. It could happen that G_01 and G_01 have different fixed points that form a 2 cycle of [G_0,G_1]. The relationship between the cycles of [F_0,F_1] and [G_0,G_1] turns out to be interesting. It has been characterized in <cit.>, but we illustrate it here for clarity and completeness.  A 2-cycle {x_0,x_1} of [F_0,F_1] must satisfy
 
    F_0(x_1,x_0)=x_0 and F_1(x_0,x_1)=x_1,   x_1≠ x_0.

A 2-cycle of [G_0,G_1] is of the form {ζ_0,ζ_1}∈ V^4× V^4 such that
G_0(ζ_0)=ζ_1 and G_1(ζ_1)=ζ_0, ζ_0≠ζ_1.
Let ζ = (x̅,y̅,u̅,v̅) be a fixed point of G_10, that
is  G_1(G_0(ζ)) = ζ.
Then we must have

    (u̅,v̅) = (F_1(y̅,x̅), F_0(x̅,y̅))  and  (x̅,y̅)=
    (F_1(v̅,u̅), F_0(u̅,v̅)).

Note that if ζ = (x̅,y̅,u̅,v̅) is a fixed point of G_10, then
ζ' = (u̅, v̅, x̅, y̅) is also a fixed point. A unique fixed point of
G_10 is therefore of the form (x̅,x̅,x̅,x̅) or
(x̅,y̅,x̅,y̅), x̅≠y̅ and satisfying the conditions (<ref>).
We discuss next the various implications of the existence of the fixed points of G_10 on
the cycles of [G_0,G_1] and [F_0,F_1].  Assume ξ_0 = (x̅,y̅,u̅,v̅) satisfies the system of equations in (<ref>).


  * (i) Let x̅=y̅. We have the following scenarios:


  * ξ̅_0=(x̅,x̅,x̅,x̅) and x̅ is a common fixed point for each individual map F_j. Obviously, x̅ will be a fixed point of [F_0,F_1].

  * ξ̅_0=(x̅,x̅,u̅,u̅), and G_0(ξ̅_0)=ξ̅_1=(u̅,u̅,x̅,x̅), where x̅≠u̅. In this case, {ξ̅_0,ξ̅_1} is a 2-cycle of [G_0,G_1] and {x̅,u̅} is a 2-cycle of the one dimensional 2-periodic system [f_0,f_1], where f_0(t)=F_0(t,t) and f_1(t)=F_1(t,t). {(x̅,x̅),(u̅,u̅)} is dubbed as a pseudo 2-cycle of [F_0,F_1].


  * (ii) Let x̅≠y̅. We have the following scenarios:


  * ξ̅_0=(x̅,y̅,y̅,x̅), where ξ̅_0 is a common fixed point of each individual map G_j. In this case, (x̅,y̅) and (y̅,x̅) are dubbed as two pseudo common fixed points of each map F_j.

  * ξ̅_0=(x̅,y̅,x̅,y̅) and G_0(ξ̅_0)=ξ̅_1=(y̅,x̅,y̅,x̅).  In this case, {ξ̅_0,ξ̅_1} is a 2-cycle of [G_0,G_1] and {y̅,x̅} is a 2-cycle of [F_0,F_1].

  * ξ̅_0=(x̅,y̅,u̅,v̅) and ξ̅_1=(v̅,u̅,y̅,x̅), where (u̅,v̅) is neither (x̅,y̅) nor (y̅,x̅).  In this case, (u̅,v̅)=(F_1(y̅,x̅),F_0(x̅,y̅)) and C_2:={ξ̅_0,ξ̅_1} is a 2-cycle of [G_1,G_0], while {(x̅,y̅),(v̅,u̅)}, {(u̅,v̅),(y̅,x̅)} are dubbed as pseudo 2-cycles of [F_0,F_1].




Next, we need to focus on stability. Given a unique fixed point of G_01, and provided we are working “in a box”, we deduce that
T_01 also has a unique fixed point. Similarly for T_10. By Lemma <ref>, [T_0,T_1] has a 2-cycle if the fixed point is not common.
This must be a globally attracting cycle <cit.>, and we extract the needed result below.


Let F_j: ℝ_+^2→ℝ_+, where F_j(↑,↓) for each j=0,1. Consider the system [F_0,F_1] with initial conditions X_0= (x_0,x_-1)∈ℝ_+^2. Suppose there exists (a,b)∈ℝ_+^2 such that a<b, (a,b)≤_se (x_k,x_k-1)≤_se(b,a) for some k≥ 0 and

    a< min{F_0(a,b), F_1(F_0(a,b),b)},  b> max{F_0(b,a),F_1(F_0(b,a),a)}.

 If G_10=G_1∘ G_0 has a unique fixed point in ℝ_+^4, then the 2-periodic system
[F_0,F_1] has a global attractor, which can be an equilibrium solution or a 2-cycle.


The condition
(a,b)≤_se (x_k,x_k-1)≤_se (b,a) means that the sequence (X_n) is eventually trapped in the box [a,b]^2. The condition (<ref>) gives us
(A,B)< G_10(A,B), where A=(a,b) and B=(b,a). Since G_10 is monotonic with respect to ≤_se, we can use our formalism directly (Section 2.1, inequalities (<ref>)) to infer that the sequence G_10^n(X_0,X_0) converges to the unique fixed point of G_10 in [a,b]^4. If the unique fixed point of G_10 is of the form ξ̅=(x̅,x̅,x̅,x̅), then x̅ is an equilibrium point of [F_0,F_1] and both {x_2n},{x_2n-1} converge to x̅. On the other hand, if the unique fixed point of G_10 is of the form ξ̅=(x̅,y̅,x̅,y̅), x̅≠y̅ then {x̅,y̅} is a 2-cycle of [F_0,F_1],  and the even terms of the orbit {x_n} converge to one branch of the 2-cycle, while the odd terms converge to the other branch.   Hence, whether the system [F_0,F_1] has an equilibrium or a 2-cycle, it serves as a global attractor.

We remark that the inequalities (<ref>) can be replaced by the more simple inequalities (a,b)<_se(F_j(a,b),F_j(b,a)), j=0,1. However, simplicity here comes at the expense of generalization.








§ STABILITY UNDER CONSTANT STOCKING


Since h>0, all solutions of Eq. (<ref>) must satisfy y_n>h for all n. Furthermore, an equilibrium solution y̅  satisfies (1-h/y̅)=e^r-y̅, and it is obvious that we obtain a unique equilibrium solution. To stress the role of the parameter h in our analysis, we denote the positive equilibrium solution by y̅_h.  The Jacobian matrix at y̅_h is given by

    J(y̅_h,y̅_h)=[
                             [       e^r-y̅_h -y̅_h e^r-y̅_h;              1              0;                ]]=[
                             [ 1-h/y̅_h   h-y̅_h;        1        0;          ]].

Let the trace of the Jacobian matrix be T and the determinant be D, then 0<T=1-h/y̅_h<1 and D=y̅_h-h>0. Furthermore, we have

    D-T=y̅_h-h+h/y̅_h-1>-1.

Based on the Jury conditions for stability, the eigenvalues of J are within the unit disk when |T|<1+D<2, and consequently, the equilibrium y̅_h is locally asymptotically stable (LAS) when D<1, i.e., y̅_h<1+h. Since

    0<e^r-y̅_h=1-h/y̅_h<1,

we obtain y̅_h>h and y̅_h > r.
In fact, y̅_0=r. The next lemma is relevant in the sequel and describes the behavior of the fixed point in terms of r and h.


    The fixed point y̅_h of Eq. (<ref>) is increasing in both h and r.



This is an immediate consequence of implicitly differentiating
(1-h/y̅_h)=d/dh e^r-y̅_h
with respect to both h and r. In the first case, we obtain

    (y̅_h-h+h/y̅_h)dy̅_h/dh=1.

So that dy̅_h/dh>0. Similarly, dy̅_h/dr > 0 and y̅_h is increasing in r.


Now, the question is: can y̅_h reach h+1 and lose its stability? If yes, then we must have h+1=e^h+1-r. This is possible when

    r=r_1:=h+1-ln(h+1).



Consider Eq. (<ref>) with h>0, and let r_1 be as defined in Eq. (<ref>).  Each of the following holds true:


  * (i) Eq. (<ref>) has a unique non-negative equilibrium solution y̅_h and y̅_h>max{r,h}.

  * (ii)  If 0<r<r_1, then y̅_h is LAS, while   if r>r_1, then y̅_h is unstable.




   Part (i) is clear. Part (ii) remains to be clarified. Recall that y̅_h is LAS if and only if y̅_h<h+1; in turn, we claim this happens iff r<r_1. But h+1 is the value of y̅_h when r=r_1, and so the claim is a consequence of the fact that y̅_h is increasing in r.  When r=r_1, the claim and its proof are standard.

When r=r_1, Eq. (<ref>) has the potential to go through a Neimark-Sacker bifurcation (see Fig. <ref>).
Note that since r_1 is increasing in h, 0<r<r_0=1 gives LAS regardless of the value of h.
Our next example illustrates the validity of Part (ii) of Lemma (<ref>). Also, Fig. <ref> shows the stability region in the (h,r)-plane.


Let h=e-1 and r=2.
Then r_1=e-1 < 2=r. In this case,
we obtain y̅_h≈ 2.898,  and the eigenvalues of the Jacobian matrix are λ≈ 0.204 ± 1.067i. They are out of the unit disk, and consequently, the equilibrium is unstable. On the other hand, if we consider h=e-1 and r=3/2, then r_1=e-1>r, y̅_h≈ 2.589 and, the eigenvalues of the Jacobian matrix are λ≈ 0.168± 0.918i. The eigenvalues are located within the unit disk; consequently, the equilibrium is LAS.



Next, we define the 4-dimensional map

    G: ℝ_+^4 →ℝ_+^4,   where   G(x,y,u,v)=(xf(y)+h,u,uf(v)+h,x).

Again here, f(y)=e^r-y. We solve G(ξ)=ξ to find the fixed points of G. We obtain ξ=(x^*,y^*,y^*,x^*), where x^*=x^*f(y^*)+h and y^*=y^*f(x^*)+h, with x^*≤ y^*. The case x^*=y^* is when both are equal y̅_h, and we write the symmetric solution ξ̅=ξ̅_h=(y̅_h,y̅_h,y̅_h,y̅_h).

Let L_1 and L_2 denote the curves of y=yf(x)+h and x=xf(y)+h, respectively. We discuss how these curves intersect since the intersection points (i.e. the “solutions”) correspond to the fixed points of G. The fixed point (y̅_h,y̅_h) is always a solution. Clearly, L_1 has a vertical asymptote at x=r and a horizontal asymptote at y=h, whereas L_2 has a vertical asymptote at x=h and a horizontal asymptote at y=r.  A variation of h makes asymmetric solutions bifurcate from the symmetric ones. The slope dy dx of L_1 at y̅_h is 1/hy̅_h(h-y̅_h), and the reciprocal of this expression is the slope of L_2. Since the slope is negative and y̅_h>h, we consider the solution of  y̅_h(y̅_h-h)=h to be

    H^*:=1/2(h+√(h^2+4h)).

In this case,

    r=r_2:=H^*+ln (H^*-h)-ln(H^*),   (and y̅_h=H^*).

Notice that we always have (we skip the computation) that

    r_2 < r_1   and  r_1<h  if  h > e-1.

To recapitulate, for a given h, and for
r=r_2, the fixed point of Eq. (<ref>)
is y̅_h = H^*, and at the solution x=y = H^*, both curves have slope -1.  The number of solutions of L_1∩ L_2 depends on whether y̅_h is larger or smaller than H^*, but this also depends on whether h<r or h>r. From a combinatorial standpoint, there are four options; however,  (h,H^*)<_se(r,y̅_h) is not viable according to the following result.


    If h≤ r, then H^*≤y̅_h.



Because 1+t≤ e^t for all t∈ℝ, then 1+h-y̅_h≤ e^h-y̅_h.
Since h≤ r, we obtain that
1+h-y̅_h≤ e^r-y̅_h,
and consequently,

    1-e^r-y̅_h≤y̅_h-h   ⟺  y̅_h(1-e^r-y̅_h)≤y̅_h(y̅_h-h) .

From the fact that y̅_h is an equilibrium point, we obtain h=y̅_h(1-e^r-y̅_h),
and therefore, h≤y̅_h(y̅_h-h), which gives us y̅_h≥ H^*.


The following lemma summarizes and settles all the cases.


Consider r_2 as defined in Eq. (<ref>). Let L_1 and L_2 denote the curves of y=F(y,x)=yf(x)+h and x=F(x,y)=xf(y)+h, respectively. Each of the following holds true:


  * (i) If h<r, then H^*<y̅_h and L_1 and L_2 intersect at the unique point (y̅_h,y̅_h).

  * (ii) If r< r_2 < h, then y̅_h<  H^* and L_1 and L_2 intersect at the unique point (y̅_h,y̅_h).

  * (iii) If r_2<r<h, then H^*< y̅_h and L_1 and L_2 intersect at three  points denoted by (y̅_h,y̅_h), (x^*,y^*) and (y^*,x^*), where x^*≠ y^*.




 (i) The two curves L_1 and L_2
always intersect at the point (y̅_h,y̅_h) along the diagonal.
Since the curves are symmetric images of one another with respect to the y=x axis, we can confine ourselves to the case
x> y̅_h> max{r,h} and investigate other intersection points. Express y explicitly for both L_1 and L_2 to find, respectively, that

    y_1 = h 1-e^r-x  and  
    y_2 = r-ln(1- h x).

We show y_2-y_1>0 for all x>y̅_h  when h<r. For a fixed value of x, we show that y_2-y_1 is positive and minimum at h=r. We have

    d dh(y_2-y_1) = -1 1-e^r-x + 1 x-h

and e^r-x> 1+r-x, for all x>r.
Therefore,
1 1-e^r-x> 1 x-r, and since  h<r<y̅_2<x,  we obtain

    d dh(y_2-y_1) < -1 x-r + 1 x-h<0.

So y_2-y_1 is decreasing and takes its minimum at h=r. At h=r,

    y_2-y_1=-r 1-e^r-x + r - ln(1-r x)>0.

 Hence, the minimum of y_2-y_1 is strictly positive, and y_2≠ y_1.  This shows that L_1 and L_2 cannot intersect again for x>y̅_h> max{r,h}, and also by symmetry, for x<y̅_h. This completes the proof of Part (i).

To show (ii) and (iii), we have shown in Lemma <ref> that the fixed point y̅_h is an increasing function of r. Therefore, if r<r_2<h, then necessarily y̅_h<H^*, since H^* is the fixed point when r=r_2. This establishes the first part of the claims in (ii) and (iii). The second part in each case follows along the same outline as in (i).


Figures <ref> and <ref> illustrate all three possible scenarios in Lemma <ref>.
















 The inequalities (x,y)≤_se (F(x,y),F(y,x)) and x≤ y have a feasible solution in case (ii) and case (iii) of Lemma <ref>, that is when h>r. This is essential for applying the embedding technique. Concerning the intersections between the curves L_1 and L_2, let g_1(t)=h_0/1-f(t),  t>r. The function g_1 is decreasing in t. Since an intersection point (a,b) must satisfy (a,b)=(g_1(b),g_1(a)), the fixed point of g_1 is the fixed point of F, and the 2-cycles of g_1 form the pseudo fixed points of F. Since a period-doubling bifurcation occurs when the fixed point of g_1 loses its stability, the existence of pseudo-fixed points can be investigated through the local stability of the fixed point of g_1.
 

After establishing the machinery and lemmas needed, we can present one of our first main results. Recall that the fixed points of the embedded map G are associated with the intersection points of L_1 and L_2. In particular, G has a unique fixed point given by ξ=ξ̅_h when cases (i) and (ii) of Lemma <ref> are satisfied, and G has three fixed points in case (iii) of that Lemma. As done earlier, we denote the asymmetric fixed points of G by ξ̅_1=(x^*,y^*,y^*,x^*) and ξ̅_2=(y^*,x^*,x^*,y^*) where  x^*<y^*. In other words, (x^*,y^*) and (y^*,x^*) are the pseudo fixed points of F.


Consider  r_2 as given in Eq. (<ref>). Each of the following holds true for Eq. (<ref>):


  * (i) If r≤ r_2, then y̅_h is globally asymptotically stable.

  * (ii) If r_2<r<h, then the compact box [x^*,y^*]^2 forms an absorbing region. In particular,

    x^*≤lim inf{x_n}≤lim sup{x_n}≤ y^*.





(i) Since r≤ r_2, Lemma <ref> guarantees a unique intersection between L_1 and L_2, and consequently, the embedding G has a unique fixed point, which we denoted by ξ̅_h = (y̅_h, y̅_h,y̅_h, y̅_h). Since r_2<r_1, y̅_h is LAS by Lemma <ref>.  To show global stability, notice that the inequalities (a,b)≤_se (F(a,b),F(b,a)) and a<b have a feasible region (as illustrated in Part (ii) of Fig. <ref> and Remark <ref>). For any initial condition (x_0,x_-1) of Eq. (<ref>), there is (a,b) in the feasible region such that Proposition <ref> is applicable. This implies that for all ξ_0=(y_0,y_-1,y_0,y_-1)  such that

    ξ=(a,b,b,a)≤_seξ_0≤_se (b,a,a,b)=ξ^t,

G^n(ξ_0) converges to ξ̅_h and Eq. (<ref>) with the initial condition (y_0,y_-1) converges to y̅_h. This verifies Part (i).
Part (ii) is similar and refers to Part (iii) of Lemma <ref> to obtain the three fixed points of G denoted by ξ̅_1,ξ̅_2 and ξ̅_h.
We have the same scenario as in Condition (<ref>) for initial conditions (x_0,x_-1) and the existence of (a,b) in the feasible region as above so that G^n(ξ_0) must converge to a fixed point of G, and by the embedding result, the orbit of Eq. (<ref>) that starts at (y_0,y_-1) will be eventually squeezed between x^* and y^* (Proposition <ref>). This completes the proof.

It is worth stressing that  r_2<r_1 always, and 0<r<r_1 guarantees the local stability of y̅_h. However, when r_2<r<min{h,r_1}, we obtain the pseudo fixed points, which cripple the embedding technique. Nevertheless, an absorbing region was obtained as given in Part (ii) of Theorem <ref>. Finally, when h<r<r_1, y̅_h is LAS. However, the absence of a feasible solution for the system (x,y)≤_se (F(x,y),F(y,x)) and x≤ y led to the unsuccessful outcome of our embedding approach for addressing global stability. Figure <ref> illustrates and summarizes our stability results in the parameter space.








§ STABILITY UNDER PERIODIC STOCKING

In this section, we consider the Ricker model with delay and p-periodic stocking as given in Eq. (<ref>), i.e., the minimal period of the stocking sequence is p. Recall that we give ourselves the liberty to use f(x) instead of exp(r-x) for writing conveniences. Define the sequence of 2-dimensional maps {T_j},

    T_j : ℝ^2_+→ℝ^2_+   as   T_j(x,y)=(xf(y)+h_j,x).

The equation X_n+1=T_n p(X_n) is just a vector form of Eq. (<ref>).
Since

    y_n+2=   y_nf(y_n)f(y_n-1)+h_0f(y_n)+h_1
    ≤    (e^r+h_0)e^r+h_1,

the composition operator T:=T_p-1∘ T_p-2∘⋯∘ T_0 maps a nonempty convex compact set into itself. Therefore, T has a fixed point in that domain. Because T_i(x,y)=T_j(x,y) if and only if i=j, then a fixed point of  T must form a p-periodic solution of X_n+1=T_n p(X_n), which reflects a p-periodic solution of Eq. (<ref>). We are interested in the global stability of the obtained p-periodic solution, and we focus on the case p=2.
The existing 2-periodic solution is in fact the 2-cycle of the system [F_0,F_1], and we denote it by {z̅_0=z̅_0(h_0,h_1),z̅_1=z̅_1(h_0,h_1)},
where

    z̅_0=z̅_1f(z̅_0)+h_1  and  z̅_1=z̅_0f(z̅_1)+h_0.

Observe that z̅_0>h_1 and z̅_1>h_0. Furthermore,  z̅_0 and z̅_1 contribute to the formation of two fixed points of T=T_1∘ T_0,  namely X̅_0:=(z̅_0,z̅_1) and X̅_1:=(z̅_1,z̅_0). Now, we give the following fact:


h_0>h_1 if and only if z̅_1>z̅_0.


From  Eqs. (<ref>), we obtain

    z̅_1-z̅_0=z̅_0(f(z̅_1))-z̅_1f(z̅_0)+h_0-h_1.

This gives us

    h_0-h_1/z̅_1-z̅_0=1-z̅_0f(z̅_1)-f(z̅_0)/z̅_1-z̅_0+f(z̅_0)>0.

Hence,   sign(h_0-h_1)=sign(z̅_1-z̅_0), which completes the proof.


Since h_0<h_1 and h_1<h_0 give similar behaviour, we focus on the case h_1<h_0. In this case, we have h_1<z̅_0< z̅_1.

 To investigate the stability of the 2-cycle, it is enough to investigate the stability of the fixed point X̅_0=(z̅_0,z̅_1) under T=T_1∘ T_0. Since

    T(x,y)=[
               [ (xf(y)+h_0)f(x)+h_1;           xf(y)+h_0;                     ]],

the Jacobian matrix at X̅_0 is

    J(X̅_0)=[
                    [ z̅_1f^'(z̅_0)+f(z̅_0)f(z̅_1)         z̅_0f(z̅_0)f^'(z̅_1);                      f(z̅_1)                z̅_0f^'(z̅_1);                              ]].

The trace and determinant here are given by

    Tr=z̅_1f^'(z̅_0)+z̅_0f^'(z̅_1)+f(z̅_0)f(z̅_1)  and   Det=z̅_0z̅_1 f^'(z̅_1)f^'(z̅_0).

We re-write the two expressions as

    Tr=   (h_1-z̅_0)+(h_0-z̅_1)+(1-h_0/z̅_1)(1-h_1/z̅_0), 
    
    Det=    (z̅_1-h_0)(z̅_0-h_1).

Obviously, Det>0 and, Det<1 when (z̅_1-h_0)(z̅_0-h_1)<1. It turns out that this condition is sufficient for the local stability of the 2-cycle as the following result shows.

 
Let h_0≠ h_1 and both are non-negative. The 2-cycle of [F_0,F_1] (i.e., {z̅_0,z̅_1}) is LAS stable if
(z̅_1-h_0)(z̅_0-h_1)<1.


We check the Jury conditions here. We have 0<Det<1. It remains to show that

    Det-Tr>-1  and   Det+Tr>-1.

The first inequality is satisfied because

    Det-Tr+1=(z̅_1-h_0+1)(z̅_0-h_1+1)-(1-h_0/z̅_1)(1-h_1/z̅_0)>0.

Now, we use this fact to write

    (z̅_1-h_0+1)(z̅_0-h_1+1)>(1-h_0/z̅_1)(1-h_1/z̅_0),

or equivalently,

    z̅_1(1-1/z̅_1-h_0)z̅_0(1-1/z̅_0-h_1)>1.

Next, write

    Det+Tr+1=   (z̅_0-h_1-1)(z̅_1-h_0-1)+(1-h_1/z_0)(1-h_0/z_1)
    
    =   (1-h_1/z_0)(1-h_0/z_1)[z̅_1(1-1/z̅_1-h_0)z̅_0(1-1/z̅_0-h_1)+1],

which is positive. This completes the proof.

Note that the condition (z̅_1-h_0)(z̅_0-h_1)<1 can be replaced by the more simple (but unnecessary) condition r≤ 1. Also, the simple conditions z̅_j ≤ h_j+1+1 for j=0,1 are sufficient to make Det<1.  For the reader's convenience, we summarize some special cases in the following corollary:

Each of the following cases is valid:


  * (i) If r≤ 1, then Det<1.

  * (ii) If z̅_j ≤ h_j+1+1 for all j=0,1, then the 2-cycle is LAS.

  * (iii) If z̅_j > h_j+1+1 for all j=0,1, then the 2-cycle is unstable.


Now, we provide some examples that illustrate the validity of Lemma <ref>.

Consider Eq. (<ref>) in each one of the following cases:


  * (i) Let h_0=2.0, h_1=6.444 and r=3.0. We obtain z̅_0≈ 6.635 and z̅_1≈ 7.237. In this case, the condition of Lemma <ref> is not satisfied. Furthermore, Det≈ 1.000 and Tr≈ -5.407. This makes Det+Tr+1<0, and consquently, the 2-cycle {z̅_0,z̅_1} of the system [F_0,F_1] is unstable. In fact, we obtain the 4-cycle

    {7.049,  19.611,  6.479,  2.000}.


  * (ii) Let r=2.0,h_0=2.156 and h_1=2.720. We obtain z̅_0≈ 3.462 and z̅_1≈ 3.199. In this case, Lemma <ref> is satisfied, and the 2-cycle {z̅_0,z̅_1} of the system [F_0,F_1] is LAS. Furthermore, Det≈ 0.774, Tr≈ -1.715 and Det+Tr+1≈ 0.059>0, and consquenlty, the eigenvalues are ≈ -0.858 ± 0.196i, which are within the unit disk.

  * (iii) Let h_0=0.820 and h_1=1.800. We let r=1.5 to obtain z̅_0≈ 2.552 and z̅_1≈ 2.151. In this case, the condition of Lemma <ref> is not satisfied. However, by considering r as a bifurcation parameter, a Neimark-Sacker bifurcation occurs near r=1.5. Indeed, the 2-cycle bifurcates into two invariant curves that serve as one attractor.  Fig. <ref> illustrates this scenario.


Based on Lemma <ref>, The 2-cycle of Eq. (<ref>) has the potential to lose its stability after going through a Neimark-Sacker bifurcation. The period-doubling observed in Part (i) of Example <ref> comes after the 2-cycle exits the stability region through the Neimark-Sacker bifurcations. Figure <ref> Shows an illustrative case of the Neimark-Sacker bifurcation that occurs.



Next, we proceed to find conditions under which we obtain global stability.  Recall F_j(x,y)=xf(y)+h_j for j=0,1. First, we investigate the existence of pseudo 2-cycles. Solving the equation G_1(G_0(ξ))=ξ, where ξ=(x,y,u,v) gives us (see Subsection 2.3)

    (x,y)=(F_1(v,u),F_0(u,v))  and   (u,v)=(F_1(y,x),F_0(x,y)).

Since each value of (x,y) determines a unique value of (u,v), it is enough to focus on the equations

    x-h_1=    (xf(y)+h_0)f(yf(x)+h_1)
    
    y-h_0=   (yf(x)+h_1)f(xf(y)+h_0).

Observe that the 2-cycle {z̅_0,z̅_1}  (see Eq. (<ref>)) satisfies System (<ref>). For other solutions, our best bargain here is to investigate the various scenarios based on the curves of the two equations. Let ℓ_1 and ℓ_2 be the curves of the first and second equations in System <ref>, respectively. The ℓ_1-curve has a horizontal asymptote at y=2r-h_1 and a vertical asymptote at x=h_1. Similarly, the ℓ_2-curve has a horizontal asymptote at y=h_0 and a vertical asymptote at x=2r-h_0. Multiple intersections between ℓ_1 and ℓ_2 guarantee the existence of artificial cycles, and in this case, the embedding technique fails to help us establish global stability. Thus, a unique intersection between ℓ_1 and ℓ_2 is crucial to our embedding strategy. Next, we appeal to the results of Subsection <ref> and Proposition <ref> in particular. Our primary objective is to identify values for (a,b) such that  a<b and the condition (<ref>) of Proposition <ref> are satisfied.
By Part (ii) and Part (iii) of Lemma <ref>, and the consequent Remark <ref>,  the inequalities (a,b)<_se (F_0(a,b),F_0(b,a)) have a feasible region when h_0>r. However, we need the feasible region to overlap with the feasible region of the inequalities

    a≤  F_1(F_0(a,b),b)  and    b≥ F_1(F_0(b,a),a).

 A quick observation here is that if (a,b)<_se (F_j(a,b),F_j(b,a)) for each j, then the inequalities in (<ref>) have a feasible region; however, we can avoid this strong constraint.  Before we proceed, we give some illustrative computational examples.





  * (i)
Consider r=1.0,h_0=2.0 and h_1=1.5. The 2-cycle is {z̅_0,z̅_1}≈{2.230,2.498}. From Eqs. (<ref>) and (<ref>), we obtain Tr≈ -1.163, and Det≈ 0.364. The eigenvalues are ≈ -0.582 ± 0.161i, which are within the unit disk. Thus, the 2-cycle {z̅_0,z̅_1} of the system [F_0,F_1] is locally asymptotically stable. The curves ℓ_1 and ℓ_2 are given in Part (i) of Fig. <ref>.

  * (ii) When r=1.0,h_0=2.0 and h_1=1.0, we obtain the 2-cycle [z̅_0,z̅_1]≈ [2.455,1.950] of the system [F_0,F_1]. The trace and determinant from Eqs. (<ref>) and
 (<ref>) are given by Tr≈ -1.42 and Det≈ -0.0732.
The eigenvalues are ≈ -1.470 and   0.050. Obviously, the 2-cycle of the system [F_0,F_1] is unstable. Next, we investigate the solution of the system G_1(G_0(ξ))=ξ. As clarified in Subsection <ref>, we explore the following scenarios:


  * G_1(ξ)=G_0(ξ)=ξ, where ξ=(x,y,y,x). x=y leads to an equilibrium solution of Eq. (<ref>), while x≠ y leads to an artificial equilibrium solution. Both scenarios are not possible since h_0≠ h_1.


  * ξ_0=(x,x,y,y), x≠ y, G_0(ξ_0)=ξ_1=(y,y,x,x) and G_1(ξ_1)=ξ_0. This means {x,y} is a 2-cycle for each one of the  one-dimensional maps f_j(x)=F_j(x,x),j=0,1. Again, this is not possible since h_0≠ h_1.


  * ξ_0=(x,y,x,y), x≠ y, G_0(ξ_0)=ξ_1=(y,x,y,x) and G_1(ξ_1)=ξ_0. This means {ξ_0,ξ_1} is a 2-cycle of [G_0,G_1] and {x,y} is a 2-cycle of [F_0,F_1]. This is always possible in our system since we verified the existence of a p-cycle for the p-periodic system [F_0,F_1,…, F_p-1]. For the specific choice of our parameters, we already computed the 2-cycle above.


  * ξ̅_0=(x̅,y̅,u̅,v̅), x̅≠y̅, (x̅,y̅)≠ (u̅,v̅), G_0(ξ̅_0)=ξ̅_1=(v̅,u̅,y̅,x̅) and G_1(ξ̅_1)=ξ̅_0.  This scenario is possible under certain choices of the parameters, and it leads to what we call artificial cycles. Indeed, when r=1.0,h_0=2.0,  and h_1=1.0, we obtain the following: The fixed points of G_10=G_1∘ G_0 are ξ̅_0=(x̅,y̅,u̅,v̅) and ξ̅_1=(u̅,v̅,x̅,y̅), where
    
    x̅≈ 1.109, y̅≈ 3.306, u̅≈ 3.966   and  v̅≈ 2.110.

    On the other hand, the fixed points of G_01=G_0∘ G_1 are η̅_0=(v̅,u̅,y̅,x̅) and η̅_1=(y̅,x̅,v̅,u̅). Therefore, [G_0,G_1] has two 2-cycles that are not generated by 2-cycles of [F_0,F_1], namely {ξ̅_0,η̅_0} and {ξ̅_1,η̅_1}. Those cycles lead to what we called artificial cycles of [F_0,F_1], namely {(x̅,y̅),(v̅,u̅)} and {(u̅,v̅),(y̅,x̅)}.
  In this case,  the embedding technique fails to tackle global stability in the system [F_0,F_1]. However, we can eventually squeeze the orbits of G_10 between ξ̅_0 and ξ̅_1 (with respect to ≤_se). Similarly, the orbits of G_01 can be eventually squeezed between η̅_0 and η̅_1 (again with respect to ≤_se). When it comes to the system [F_0,F_1], the even terms of the orbits are eventually squeezed between x̅ and u̅, while the odd terms of the orbits are eventually squeezed between v̅ and y̅.











 Now, we revisit the system of inequalities in (<ref>), and re-write them as
 
    a(1-(f(b)))^2≤    h_0f(b)+h_1
    
     b(1-(f(a)))^2≤    h_0f(a)+h_1.

 For a,b>r, the factors on the left-hand side are positive, which motivate us to define the map g_2(t)=h_0f(t)+h_1/1-(f(t))^2 ,  t>r.  Also, recall from Remark <ref> that we defined the map g_1(t)=h_0/1-f(t). Both g_1 and g_2 are decreasing in t. The inequalities in (<ref>) become a≤ g_2(b) and b≥ g_2(a). It is straightforward to check the following result.
 
 Assume h_0>r. There exists an unbounded set of points (a,b),  a<b that satisfy
 
    a≤ g_1(b)<g_2(b)<g_2(a)  whenever   h_0<h_1

 and
 
    b≥ g_1(a)>g_2(a)>g_2(b)  whenever   h_0>h_1.

 


 Finally, we arrive at the main result of this section, which provides a generalization of Part (i) of Theorem <ref>.


Consider Eq. (<ref>) with h_0≠ h_1. If both h_0,h_1>r and the solution of System (<ref>) is unique, then the 2-cycle {z̅_0,z̅_1} is global asymptotically stable.



For each initial condition (x_0,x_-1) of Eq. (<ref>), we need to find a point (a,b) that makes Proposition <ref> applicable. Without loss of generality, we can consider the initial conditions to be (x_2,x_1). In this case, we guarantee that x_1>h_0>r and x_2>h_1>r.  If h_0>h_1, we consider (a,b) such that a<b, b=g_1(a) and a is sufficiently small so that r<a<h_1. In this case and based on Lemma  <ref>, we obtain

    b≥ g_1(a)>g_2(a)>g_2(b)>h_1>a.

If h_0<h_1, we consider (a,b) such that a<b, a=g_1(b) and b is sufficiently large so that g_2(h_0)<b. In this case and based on Lemma  <ref>, we obtain

    a≤ g_1(b)<g_2(b)<g_2(a)=g_2(g_1(b))<g_2(h_0)<b.

Observe that this choice of (a,b) can be done to have (x_2,x_1)∈ [a,b]^2. All the hypotheses of Proposition <ref> are satisfied, and we invoke the Lemma to obtain the result.

The uniqueness condition in Theorem <ref> is to guarantee that the global attractor is the 2-cycle of [F_0,F_1]. However, if we require h_0,h_1>r, then the theorem gives us either a globally attracting 2-cycle or a trapping box in which its boundaries are determined by two artificial 2-cycles of [F_0,F_1].



§ CONCLUSION

 We investigated two systems in this paper: the Ricker model with a time delay and constant stocking x_n+1=F(x_n,x_n-1)=x_nexp(r-x_n-1)+h, and the system involving periodic stocking x_n+1=F_n(x_n,x_n-1)=x_nexp(r-x_n-1)+h_n.  In both cases, we embedded the 2-dimensional system into a 4-dimensional monotonic system with respect to a partially ordered set. A unique positive equilibrium exists in the first system, which we denote by y̅_h. The equilibrium y̅_h transitions into a cycle in the second system, and the cycle inherits the system's period. We utilized the embedding technique to analyze global stability in both systems.
  


  In the constant stocking case, we obtained local stability when y̅_h<1+h, which leads to the condition 0<r<r_1=h+1-ln(h+1). On the other hand, we proved global stability (see Theorem <ref>) under the constraint

    y̅_h<1/2(h+√(h^2+4h)),

which leads to the condition

    0<r<r_2=1/2(h+√(h^2+4h))+ln(1-2h/h+√(h^2+4h)).

It's worth noting that the global stability condition we established is more stringent than the local stability condition, i.e., r_2<r_1. Therefore, the common assertion that “local stability implies global stability" is still an open problem for all values of r between r_1 and r_2 as we illustrated in Fig. <ref>.



For the 2-periodic system x_n+1=F_n 2(x_n,x_n-1), we established conditions that ensure both local and global stability of the existing 2-cycle (see Theorem <ref>). We illustrated a scenario in which the 2-cycle undergoes a Neimark-Sacker bifurcation, leading to the emergence of two invariant curves that act as a unified attractor (see Example <ref>). Finding the explicit form of the cycle is not possible, but we provided illustrative numerical examples that clarify various possible scenarios (see Example <ref>).
As in the constant stocking case, sufficient conditions for the global stability of the 2-cycle are obtained in the 2-periodic case. Nonetheless, identifying necessary and sufficient conditions remains an open problem. We also conjecture here that the local stability of the 2-cycle is sufficient to ensure its global stability.




Acknowledgment: The authors express their gratitude to the anonymous reviewers for their valuable comments. This work does not have any conflicts of interest. There are no funders to report for this submission. 




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