Kneading sequences for toy models of Hénon maps
Ermerson Araujo






Kneading sequences for toy models of Hénon maps]Kneading sequences for toy 
 models of Hénon maps





[2010]37B10, 37E05

Ermerson Araujo, Departamento de Matemática, Centro de Ciências, Campus do Pici,
Universidade Federal do Ceará (UFC), Fortaleza – CE, CEP 60440-900, Brasil
ermersonaraujo@gmail.com











The purpose of this article is to study the relation between combinatorial equivalence and 
topological conjugacy, specifically how a certain type of combinatorial equivalence implies 
topological conjugacy. We introduce the concept of kneading 
sequences for a setting that is more general than one-dimensional dynamics:
for the two-dimensional toy model family of  Hénon maps introduced by Benedicks and Carleson, 
we define kneading sequences for their critical lines, and 
prove that these sequences are a complete invariant
for a natural conjugacy class among the toy model family.
We also establish a version of Singer's Theorem for the toy model family.


[
    Ermerson Araujo
    December 30, 2023
=====================












§ INTRODUCTION


One of the main questions in dynamical systems is to identify when two
systems are `the same', where the term `the same' means 
some type of equivalence between the systems. 
One of the simplest notions of equivalence is topological conjugacy.
Let X and Y be topological spaces, and let f:X→ X and g:Y→ Y be 
continuous maps. We say that f,g are topologically 
conjugate (or simply conjugate) if there is a 
homeomorphism h:X→ Y satisfying the conjugacy equation h∘ f=g∘ h. 

Topological conjugacy heavily depends on class of maps. For orientation preserving
circle homeomorphisms, Poincaré introduced in the late XIX century the notion of rotation numbers,
which essentially characterises all conjugacy classes: if f:𝕊^1→𝕊^1
is an orientation preserving 
homeomorphism with irrational rotation number ρ, then f is topologically semiconjugate to
the rigid rotation R_ρ(x)=x+ρ.
Furthermore, if f contains a dense orbit then f is indeed topologically conjugate to R_ρ.

For interval homeomorphisms, topological conjugacy is 
trivial, due to the following fact: every orbit is either periodic 
or converges to a periodic orbit. Now, if we consider interval 
endomorphisms, then the situation is much richer.
For this class, Milnor and Thurston developed a theory that substitutes the theory of rotation
numbers of Poincaré <cit.>. Nowadays known as kneading theory, it 
plays an important role in one-dimensional dynamics. It allows, for example,
to prove that topological entropy varies continuously in the the quadratic family, see <cit.> for details.
The setting considered by Milnor and Thurston is the following: let I be an interval,
and let f:I→ I be a map with finitely many turning points (i.e. a point x∈ I where
f changes monotonicity). For each turning point, they defined a sequence, called kneading sequence, 
that encodes the trajectory of this point. Then they proved that if two interval maps do not have wandering intervals,
periodic attractors neither intervals of periodic points, then they are topologically conjugate 
if and only if their kneading sequences are the same. 

When we pass to two-dimensional maps, the situation becomes much more complicated.
One of the main maps considered is H_a,b:ℝ^2→ℝ^2 given by
H_a,b(x,y)=(a-x^2-by,x), a,b∈ℝ, known as Hénon map.
Over the last years, there has been various attempts to construct a similar theory in this context,
but none of them is yet completely satisfactory. The difficulty comes from two facts: the plane does not possess
a natural order as the interval, and there is no sufficiently nice dynamical notion of critical point. 

In this work we introduce kneading sequences for a particular two-dimensional family.
This family was considered by Benedicks and Carleson as a toy model for the study of Hénon maps <cit.>.
Each of these maps acts on a two-dimensional rectangle via the expression F(x,y)=(f(x,y), K(x,y)),
where f is a family of unimodal maps and K is an inverse branch of a Cantor map. We call 
each map of this form a toy model.
Matheus, Moreira and Pujals proved that, among the toy models, Smale's Axiom A property is 
C^1-dense and, on the C^2-topology, there exists an open subset satisfying a
Newhouse-like phenomenon <cit.>. These properties indicate that the toy models exhibit 
rich dynamical properties. It is for them that we characterize topological conjugacy in terms
of kneading sequences.


Let F,G be two toy models, and assume that
neither of them has wandering intervals, interval of periodic points nor
weakly attracting periodic points. Then F,G are fiber topologically conjugate 
if and only if they have the same kneading sequences.


The hardest part in Theorem <ref> is the reverse implication, that
same kneading sequences lead to fiber topological conjugacy. To construct the conjugacy,
we employ ideas similar to those used in 
one-dimensional dynamics: we define it in preimages of turning points and
then extend to the closure. This approach also leads to the same result for more general toy models,
see Theorem <ref>.

Theorem <ref> has an underlying motivation that we now explain.
For dissipative surface diffeomorphisms, Crovisier and Pujals introduced 
a new class of diffeomorphisms, called strongly dissipative, whose 
dynamical properties behave as an intermediate level between one-dimensional dynamics
and two-dimensional dynamics <cit.>. They proved that this class is open and non-empty (it contains 
Hénon maps with Jacobian in (-1/4, 1/4)), and that every strongly dissipative diffeomorphism of the disc 
(in particular Hénon maps) is equivalent to a one-dimensional structure (it is semiconjugate to an 
endomorphism defined on a tree). If two strongly dissipative diffeomorphisms on the disc are topologically
conjugate, then their one-dimensional structures are also conjugate. It is unknown
if this one-dimensional structure carries all relevant dynamical information, i.e., if the 
one-dimensional dynamics are conjugate, are the strongly dissipative diffeomorphisms also conjugate?
Theorem <ref> shows that, on the
presence of a good definition of turning points, we can recover information about the conjugacy class
of the toy model by means of an appropriate combinatorial structure. We hope that the methods
introduced in the proof of Theorem <ref> will be used to study the conjugacy 
classes of strongly dissipative diffeomorphisms of the disc. 

It is worth mentioning that kneading sequences are not the only method to characterize
topological conjugacy. Based on numerical experiments <cit.>, Cvitanović introduced 
in <cit.> the concept of pruning fronts and conjectured that every map 
H_a,b in the Hénon family 
can be understood as a pruned horseshoe: 
if F:ℝ^2→ℝ^2 is Smale's horseshoe map, 
then after pruning (destroying) some orbits of F we construct 
a new map F that is equivalent, in some sense, to H_a,b. 
This is known as the Pruning Front Conjecture. 
Mendonza proved that the Pruning Front Conjecture holds in an open set of
the parameter space <cit.>, and Ishii proved it for the Lozi family <cit.>.
For more detail on the theory of pruning fronts and its relationship with kneading theory,
see <cit.>. 

On a different direction of kneading sequences, we also establish for the toy models
a version of the classical Singer theorem. Let us explain this. Singer proved that, 
for C^3 interval maps with negative Schwarzian derivative, the basin of any
attracting periodic point contains either a critical point or a boundary point of the interval.
Using that the toy models do preserve horizontal lines, 
we prove the following theorem, which is a version of Singer's theorem for the toy models.


Let F(x,y)=(f(x,y), K(x,y)) be a toy model.
If each interval map f(·,y) has negative Schwarzian 
derivative, then the closure of the immediate 
basin of any strongly attracting periodic orbit contains either a 
point of the critical line or a boundary point of the rectangle.


This paper is organized as follows. Section <ref> introduces in full details the 
toy models and consider their kneading sequences. It also states the main results, establishes 
some basic properties of the toy models, and discusses examples.
Section <ref> contains the proofs of the main results stated in Section <ref>.


 

Acknowledgements. This article is part of my Ph.D thesis at IMPA
under the supervision of Enrique Pujals. I deeply thank him for his constant
support, patience, encouragement and availability. I also would like to thank
Yuri Lima for the carefully reading and several helpful comments. This
work was supported by CNPq and CAPES.










§ TOY MODELS AND KNEADING SEQUENCES


Results in one-dimensional dynamics heavily rely on the total order of the line.
For example, given two multimodal interval maps, the preimages
of the turning points divide the interval from left to right, and if these preimages
are combinatorially equivalent then there is a monotone bijection
between them. The family of two-dimensional maps we consider also inherits this property:
as we will see below, the one-dimensional order structure
is preserved along “unstable leaves”.










 §.§ The Toy Model


Given a continuous interval map f:[a,b]→ [a,b], we call it
a unimodal map if f(a)=f(b)=a and if there exists a
unique point c≠ a,b, called turning point, for which f is
increasing on [a,c] and decreasing on [c,b]. We consider two-dimensional maps 
with fiber dynamics given by unimodal maps. We follow <cit.> for the description.

One-Parameter Family of Unimodal Maps: Consider a one-parameter family

    f(y):[-1,1]→[-1,1],  y∈[0, 1],

such that:


  * y↦ f(y) is continuous,

  * each f(y) is a unimodal map with f(y)(-1)=f(y)(1)=-1 and 0 as a turning point.




Fix a<b in (0,1).

Cantor Map: A Cantor map is a differentiable map k:[0,a]∪[b,1]→[0,1] such that
k(0)=k(1)=0, k(a)=k(b)=1 and | k'|>γ>1.
Each Cantor map defines a Cantor set 𝒦^F given by

    𝒦^F=⋂_n≥0k^-n([0,a]∪[b,1]).


Let K_+=(k_[0, a])^-1 and K_-=(k_[b, 1])^-1 be the inverse branches of k,
and define K(x,y) by

    K(x,y)= {[  K_+(y),  if  x>0;  K_-(y), if  x<0. ].


Toy Model: A toy model is a map F of the form

    [                     F  :      ([-1,1]\{0})× [0,1]                        ⟶             [-1,1]×[0,1];                                             (x,y)                        ⟼ (f(y)(x), K_sign(x)(y)). ]




The map F is not defined on the vertical line {0}× [-1,1], and this 
may introduce serious difficulties to analyse F and apply compactness arguments.
To bypass this, we duplicate each (0,y) to two points (0^-,y) and (0^+,y)
and define F on them by F(0^±,y)=(f(y)(0),K_±(y)).

Critical Line of F: It is the set 

    _c(F)={(0^±, y):y∈[0,1]}
 
equal to the disjoint union of two vertical segments {0^-}× [0,1] and {0^+}× [0,1].
Each element of _c(F) is called a turning point of F.

Therefore F is now defined on (([-1,1]\{0})×[0,1])∪_c(F),
call this set Dom(F). We define a natural topology on Dom(F) as follows:


  * If (x,y)∈Dom(F)\_c(F), then 
the neighborhoods of (x,y) are the standard Euclidean neighborhoods.

  * If x=0^- then the neighborhoods
of (x,y) are 𝒰∩ ([-1,0^-]× [0,1]), where 𝒰 is  a standard
Euclidean neighborhood of (0,y).

  * If x=0^+ then the neighborhoods
of (x,y) are 𝒰∩ ([0^+,1]× [0,1]), where 𝒰 is  a standard
Euclidean neighborhood of (0,y).

We remark that, although F is now defined on a compact set, it is not continuous.










 §.§ Notations and preliminaries


For y∈[0,1], write f_-(y) for the restriction f(y)_[-1, 0^-] and
f_+(y) for the restriction f(y)_[0^+, 1], see Figure <ref>. 


Also, let F_- and F_+ be the restrictions of F defined by

    [                   F_-  :          [-1,0^-]× [0,1]                        ⟶ ([-1,0^-]∪(0^+,1])×[0,1];                                             (x,y)                        ⟼      (f_-(y)(x), K_-(y)) ]

and

    [                   F_+  :           [0^+,1]× [0,1]                        ⟶ ([-1,0^-)∪[0^+,1])×[0,1];                                             (x,y)                        ⟼     (f_+(y)(x), K_+(y)). ]

Every orbit of F is therefore obtained by compositions of F_- and F_+:
for each (x,y)∈Dom(F) there is a sequence 
j(x,y)=(j_1j_2⋯ j_m⋯)∈{-,+}^
such that

    F^m+1(x,y)=F_j_m+1(x_m,y_m),  m≥ 0,

where 

    y_m=(K_j_m∘⋯∘ K_j_1)(y)=:y_j_m⋯ j_1 and 
    x_m=f_j_m(y_j_m-1⋯ j_1)∘⋯∘
    f_j_2(y_j_1)∘ f_j_1(y)(x).

Let I(y):=([-1,0^-]∪[0^+,1])×{y}, and
call J⊂ I(y) an interval if there are
x_1,x_2∈[-1,0^-] or x_1,x_2∈[0^+,1] such that

    J={x∈[-1,0^-]∪[0^+,1] :  x_1< x< x_2}×{y}.

We will sometimes denote J as above by [(x_1,x_2),y]. If 

    J={(x,y)∈[-1,0^-]∪[0^+,1] :  x_1≤ x≤ x_2}×{y},

then we write J=[[x_1,x_2],y]. 
Given a periodic point (p,q) of F, set

    ℬ(p,q)={(x,y): F^ℓ(x,y)→𝒪(p,q) as ℓ→∞}.

We call (p,q) a weakly attracting periodic point if
ℬ(p,q) contains an interval. If ℬ(p,q)
contains an open subset, then call (p,q) a
strongly attracting periodic point. 
When 𝒪_F(p,q)∩_c(F)=∅, then
the immediate basin ℬ_0(p,q) of 𝒪(p,q) is the union of the 
connected components of ℬ(p,q) that contain of the set
from {(p,q), F(p,q),…, F^n-1(p,q)}, 
where n is the period of (p,q). Hence ℬ_0(p,q)∩_c(F)=∅,
and so there is a sequence j_1⋯ j_n∈{-,+}^n
such that for each (x,y)∈ B(p,q) we have

    (f_j_n(y_j_n-1⋯ j_1)∘⋯∘
    f_j_2(y_j_1)∘ f_j_1(y)(x), 
    y_j_n⋯ j_1)∈ B(p,q),

where B(p,q)⊂ℬ_0(p,q) is the
component containing (p,q).
Clearly, if (p,q) is a strongly attracting 
periodic point, then it is also a weakly attracting periodic point. 

Given y∈[0,1], an interval J⊂ I(y) is called an
interval of periodic points if every (x,y)∈ J is a
periodic point of F.

Wandering Interval: An interval J⊂ Dom(F) is called
wandering if 


  * F^n(J)∩ F^m(J)=∅, ∀ n≠ m;

  * F^n(J)∩_c(F)=∅, ∀ n≥ 0.




A point (x,y) is called non-wandering if
every neighborhood V⊂Dom(F) containing (x,y) 
has an iterate ℓ≥1 such that F^ℓ(V)∩ V≠∅.
The non-wandering set Ω(F) is the set of all non-wandering points. 

It is well known that if f:I→ I is an unimodal map without
intervals of periodic points nor attracting periodic points,
then the set of pre-images of the turning point is dense in Ω(F).
We establish the same for toy models. Define

    (F)={(x,y)∈Dom(F):∃ ℓ≥0 
     such that  F^ℓ(x,y)∈_c(F)}.



Let F be a toy model, and assume that F has no weakly attracting periodic point nor intervals of
periodic points. Then Ω(F)⊂(F).



The proof is by contradiction.
Suppose that (x,y)∈Ω(F)\(F).
Let V be a neighborhood of (x,y) such that 
F^n(V)∩_c(F)=∅ for all n≥0.
Without loss of generality, we can assume that V=I× J, 
where x∈ I, y∈ J, and I,J are open intervals. 
Fix m≥1 with F^m(V)∩ V≠∅, and put 

    L=⋃_n≥0 F^n(V).

Since, for each r=0,…,m-1, the union 
⋃_j≥0 F^jm+r(V) is connected, the set
L has finitely many connected components.
Set L=L_1⊔⋯⊔ L_s, where L_1,…,L_s are connected.
Using that F(L)⊂L, for each i 
there exists j such that F(L_i)⊂ L_j. 
Let L be the connected component containing (x,y).
We have F^m(L)∩ L≠∅, hence F^m(L)⊂ L.
Take ℓ=min{d≥1; F^d(L)⊂ L}.
Hence, we have that L is a domain such that F^ℓ(L)⊂ L,
F^i(L)∩_c(F)=∅ for i=0,1,…,ℓ-1, and 
F^i(L)∩ F^j(L)=∅
for i,j=0,1,…ℓ-1 with i≠ j. 
Therefore there exists a sequence j_0j_1⋯ j_ℓ-1∈{-,+}^ℓ
such that for any (a,b)∈ L and n≥0 
it holds F^nℓ(a,b)=(a_nℓ,b_(j_ℓ-1⋯ j_1j_0)^n), where
b_(j_ℓ-1⋯ j_1j_0)^n= (K_j_ℓ-1∘⋯∘ K_j_0)^n(b)
and 

    a_nℓ=f_j_ℓ-1(b_j_ℓ-2⋯ j_0(j_ℓ-1⋯ j_0)^n-1)
    ∘⋯∘
    f_j_ℓ-1(b_j_ℓ-2⋯ j_0)∘⋯∘ 
    f_j_1(b_j_0)∘ f_j_0(b)(a).
 
Let π_2 be the projection on the second coordinate, π_2(x,y)=y.
Since K:=K_j_ℓ-1∘⋯∘ K_j_0 
is a contraction, there exists w∈π_2(L) such that K(w)=w.
We claim that w=y. To see that,
for each n≥1 take V_n:=I_n× J_n I× J 
neighborhood of (x,y) such that 
I_n+1× J_n+1 I_n× J_n  with 
|I_n|→0 and |J_n|→0 as n→∞.
As before, for each n≥1 there are
(x_n,y_n)∈ V_n and k_n≥1 such that 
F^k_n(x_n,y_n)∈ V_n and 
F^k_n(x_n,y_n)→(x,y) as n→∞. 
We may suppose that k_n→∞ as n→∞.
Furthermore, by the construction of L, 
we have k_n=r_nℓ for some r_n≥1.
We already know that K is a contraction, 
hence there is λ<1 such that 

    |K(x)-K(y)|≤λ|x-y|, ∀ x,y∈[0,1].

Note that

    |K^r_n(y_n)-w|   ≤   |K^r_n(y_n)-K^r_n(y)|+|K^r_n(y)-w| 
       ≤   λ^r_n|y_n-y|+|K^r_n(y)-w|→ 0,

therefore

    π_2[F^k_n(x_n,y_n)]=K^r_n(y_n)→ w.

Using that F^k_n(x_n,y_n)→(x,y) as n→∞, we conclude that w=y, as claimed.

Now, for every (v,y)∈ L∩ I(y) and n≥1 we get

    F^mℓ(v,y)=((f_j_ℓ-1(y_j_ℓ-2⋯ j_0)
    ∘⋯∘ f_j_0(y))^m(v),y)∈ L∩ I(y),

where f_j_ℓ-1(y_j_ℓ-2⋯ j_0)
∘⋯∘ f_j_0(y) is a strictly monotone map. 
Without loss of generality, we assume that
f_j_ℓ-1(y_j_ℓ-2⋯ j_0)∘⋯∘ f_j_0(y) is strictly increasing.
Let {I_i}_i be the connected components of L∩ I(y).

Claim: There exists i_0 such that F^ℓ(I_i_0)⊂ I_i_0.

Proof of the claim. Assume, by contradiction, that it does not hold.
Since the composition f_j_ℓ-1(y_j_ℓ-2⋯ j_0)∘⋯∘ f_j_0(y) 
is strictly increasing, for every i there exists σ(i) such that
F^ℓ(I_i)⊂ I_σ(i) and sup I_i<inf I_σ(i).
Fixed any k, this implies that the sequence {F^ℓ n(I_k)}_n≥0
is formed by disjoint intervals satisfying sup F^ℓ n(I_k)<
inf F^ℓ(n+1)(I_k) and |F^ℓ n(I_k)|→0 as n goes to infinity.
This implies that there is (v_0,y) such that F^ℓ(v_0,y)=(v_0,y) and
F^ℓ n(I_k)→(v_0,y), contradicting the non-existence of 
weakly attracting periodic point. The claim is proved.

Now we complete the proof of the proposition. Let i_0 be given by the claim. The
map F^ℓ:I_i_0→ I_i_0 is strictly increasing, hence either I_i_0 contains 
an interval of periodic points for F^ℓ , or some open subinterval 
of I_i_0 converges to a single periodic point. Both possibilities contradict the 
underlying assumptions on F.











 §.§ Kneading sequences and main results


Let π_1 be the projection on first coordinate, π_1(x,y)=x. Let F be a toy model. 
Consider the alphabet 𝒜={L, 0^-,0^+, R}. The letter L means “left”
and R means “right”.

Address of a point: The address of a point (x,y)∈Dom(F) is 
the letter t_F(x,y)∈𝒜 defined by:

    t_F(x,y)= {[                 L   , if π_1(x,y)<0;               0^- , if π_1(x,y)=0^-;               0^+ , if π_1(x,y)=0^+;                 R  , if π_1(x,y)>0. ].



Itinerary of a point: The itinerary of a point (x,y)∈Dom(F)
is the sequence T_F(x,y)∈𝒜^{0,1,2,…} defined by 

    T_F(x,y)=(t_F(x,y),t_F(F(x,y)),…,t_F(F^n(x,y)),…).


Kneading Sequences: The kneading sequences of F are all the itineraries
of elements of _c(F). We denote this set of itineraries by 𝒦_F(_c). 

In the following proposition, we construct conjugacies between the Cantor sets appearing in 
the second coordinate of two toy models. The proof is standard, but we include it for completeness.


 
Let F(x,y)=(f(y)(x), K^F_ sign(x)(y)) and 
G(x,y)=(g(y)(x),
 K^G_ sign(x)(y)) 
be two toy models. There exists a strictly increasing continuous 
map ψ:[0,1]→[0,1] such that  

    ψ∘ K^F_±=K^G_±∘ψ.
 
Moreover, ψ_𝒦^F is independent of the choice of ψ.



Let k^F:[0,a^F]∪[b^F,1]→[0,1] and k^G:[0,a^G]∪[b^G,1]→[0,1]
the Cantor maps of F and G.
If 𝒦^F and 𝒦^G are the Cantor sets generated by k^F and k^G respectively, then
there is a unique strictly increasing bijection 
ψ:𝒦^F→𝒦^G such that ψ∘ k^F=k^G∘ψ. To see this,
observe that if we order {0,1}^ lexicographically, then 𝒦^F and 𝒦^G
are isomorphic to {0,1}^ through an order preserving bijection. 
Hence, for each x∈𝒦^F we take ψ(x) to be
the unique point of 𝒦^G with the same symbolic representation
as x in 𝒦^F.
Now we extend ψ continuously to gaps of 𝒦^F. Take any 
strictly increasing homeomorphism h:(a^F, b^F)→(a^G, b^G).
Denote by k^F,G_+ the restriction k^F,G_[0, a^F,G] and
by k^F,G_- the restriction k^F,G_[b^F,G, 1].
Given a gap J⊂[0,1] of 𝒦^F, there exists a unique n≥1 and 
a sequence j_1⋯ j_n∈{-,+}^n such that
(k^F_j_n∘⋯∘ k^F_j_1)(J)=(a^F, b^F). 
Thus, for each x∈ J we define ψ(x) by

    ψ(x):=[(k^G_j_1)^-1∘⋯∘ (k^G_j_n)^-1]
    ∘ h∘ [k^F_j_n∘⋯∘ k^F_j_1](x).

Clearly ψ:[0,1]→[0,1] is a strictly increasing continuous 
map with ψ∘ K^F_±=K^G_±∘ψ.
By construction, ψ_𝒦^F does not depend on the choice of ψ.   



Consider two toy models F(x,y)=(f(y)(x), K^F_ sign(x)(y)) and 
G(x,y)=(g(y)(x),K^G_ sign(x)(y)), and let ψ:[0,1]→[0,1] be the 
map given by Proposition <ref>. 
We say that F and G have the same kneading sequences, and 
write 𝒦_F(_c)=𝒦_G(_c),
if 

    T_F(0^±,y)=T_G(0^±, ψ(y)), ∀ y∈[0,1].



Observe that the above definition does not actually depend on the choice of ψ.
Indeed, the invariant set defined by F is 
([-1,0^-]∪[0^+,1])×𝒦^F, as we know that in 𝒦^F
the map ψ is uniquely defined. We are now ready to state 
the main tool to be used in the proof of Theorem A.


Let F, G be two toy models, and assume that 𝒦_F(_c)=𝒦_G(_c).
Then there exists a bijective map 
H:(F)→(G) such that:


  (i) H∘ F=G∘ H on (F)\_c(F).

  (ii) H((F)∩ I(y))⊂(G)∩ I(ψ(y)) for all y∈[0,1].

  (iii) H_(F)∩ I(y) is strictly increasing for all y∈[0,1].



The proof of Theorem <ref> will be given in the next section.
When the map H given by Theorem <ref> exists, we say that
F and G are combinatorially equivalent.
 

Consider two toy models F(x,y)=(f(y)(x), K^F_ sign(x)(y)) and 
G(x,y)=(g(y)(x),K^G_ sign(x)(y)), and let ψ:[0,1]→[0,1] be the 
map given by Proposition <ref>.
We say that F and G are fiber topologically semiconjugate 
if there is a continuous surjective map H: Dom(F)→ Dom(G)
which is increasing on intervals, such that
H∘ F=G∘ H and H(I(y))⊂ I(ψ(y)) for all y∈ [0,1].
If H is a homeomorphism, then we say that F and G are fiber topologically conjugate.


If H conjugates F and G as above, then H(0^±,y)=(0^±,ψ(y)) for all y∈[0,1].
In particular, H(_c(F))=_c(G).
The following proposition states that kneading sequences are a topological invariant
for the notion introduced in Definition <ref>.



Let F, G be two toy models. If F,G are fiber
topologically conjugate then 𝒦_F(_c)=𝒦_G(_c).



Let H: Dom(F)→ Dom(G) be conjugacy between F and G.
We will show that T_F(0^±,y)=T_G(0^±,ψ(y)) for all y∈[0,1].
Fix y∈[0,1].
Clearly t_F(0^±,y)=t_G(0^±,ψ(y)).
Since F,G are fiber topologically conjugate,

    H[F^n(0^±,y)]=G^n[0^±,ψ(y)], ∀ n≥ 1.

Since H is strictly increasing on intervals,

    t_F(F^n(0^±,y))=t_G(G^n[0^±,ψ(y)]), ∀ n≥ 1.

Therefore T_F(0^±,y)=T_G(0^±,ψ(y)). 



Notice that the above proposition implies the first part of Theorem <ref>.
Moreover, within the notion of topological conjugacy given by Definition <ref>,
Theorem <ref> establishes that kneading sequences are a complete topological invariant.
It would be natural to find, inside the class of toy models,
a family of representatives for each  combinatorial equivalence class. 
To better understand this question, we remind a classical result
in one-dimensional dynamics. Consider the quadratic family {f_μ}_0≤μ≤ 4
of maps f_μ:[0,1]→[0,1] defined by f_μ(x)=μ x(1-x).
This family is universal for unimodal maps in the following sense: if g:[0,1]→[0,1]
is a unimodal map then g is combinatorially equivalent to f_μ for some μ=μ(g).
Using this analogy, it is natural to find a universal family for toy models. 
At the moment, it is not clear how to construct such a family, left alone its existence. 



Let F be a toy model.

Wandering Domain: A domain V⊂ Dom(F) is called wandering if 


  * F^n(V)∩ F^m(V)=∅, ∀ n≠ m;

  * F^n(V)∩_c(F)=∅, ∀ n≥ 0.



 
For y∈[0,1] and n≥1, let

    _n^F(y):={(x,y)∈Dom(F)∩ I(y):
     F^ℓ(x,y)∈_c(F) 
     for some  0≤ℓ≤ n-1}.

Consider the map ϕ_n^F:[0,1]→2^Dom(F) defined by 
ϕ_n^F(y):=_n^F(y). Under some regularity of this function,
we can state a weaker version of Theorem <ref>.


Let F,G be two toy models. Assume that
F,G have no wandering domain and that 𝒦_F(_c)=𝒦_G(_c). 
If the family {ϕ_n^G}_n≥ 1 is equicontinuous
then F,G are fiber topologically semiconjugate.


Above, equicontinuity means the following: given ε>0 there is 
δ>0 such that for all y_1,y_2∈[0,1] with |y_1-y_2|<δ it holds 
d_ℋ(_n^G(y_1),_n^G(y_2))<ε for all n≥1,
where d_ℋ is the Hausdorff distance. 
At the moment, we observe that the assumptions that G has no wandering domain and that
{ϕ_n^G} is equicontinuous imply that G has no wandering intervals, therefore
Theorem <ref> follows from Theorem <ref>. Nevertheless,
we will give an independent proof Theorem <ref> in the next section.



 §.§ Examples


We now give some examples of toy models, and discuss the
hypotheses required in Theorem <ref> and Theorem <ref>. 


Let f:[-1,1]→[-1,1] be the tent map, i.e.
f(x)=1+2x for x∈ [-1,0] and f(x)=1-2x for x∈[0,1]. Let k:[0,1]→[0,1] be the Cantor 
map defined by k(y)=3y if y∈[0,1/3] and k(y)=3-3y if y∈[2/3,1]. 
The map F(x,y):=(f(x), K(x,y)) satisfies the hypotheses of Theorems <ref>
and <ref>.



Let f:[-1,1]→[-1,1] be a unimodal map with 0 as turning point and containing
a wandering interval I_0. Let k be the Cantor map of example 1. 
The toy model F(x,y):=(f(x), K(x,y)) has wandering intervals I_0×{y}.



Let q,f,g:[-1,1]→[-1,1] be three unimodal maps defined by

    q(x)=-x^2,  
    f(x)= 
    {[         x , if  x≤0;        -x , if  x≥0 ].,
      g(x)= {[     -√(-x)  , if  x≤0;      -√(x) , if  x≥0. ].



Let k be the Cantor map defined in example 1, and define the toy models
Q(x,y)=(q(x), K(x,y)), F(x,y)=(f(x), K(x,y)), and G(x,y)=(g(x), K(x,y)). Then

    𝒦_Q(_c)=𝒦_F(_c)=𝒦_G(_c).

By Theorem <ref>, there exists H_Q,F:𝒞(Q)→𝒞(F),
H_F,G:𝒞(F)→𝒞(G) and H_G,Q:𝒞(G)→𝒞(Q).
On the other hand, they are not conjugate 
since the turning point of q,f,g is respectively an attracting, 
neutral and repelling fixed point. 







 §.§ Generalized toy models


In this section we introduce a larger class of toy models, changing unimodal to multimodal maps.
We say that an interval map f:[0,1]→[0,1] is an ℓ-modal map if it has 
exactly ℓ turning points c_1<⋯<c_ℓ. This means that 
the intervals [0,c_1],[c_1,c_2],…,[c_ℓ,1] are the largest intervals in which f is monotone.
Fixed  c_1<⋯<c_ℓ, consider a one-parameter family

    f(y):[0,1]→[0,1], y∈[0,1],

of ℓ-modal maps with turning points c_1<⋯<c_ℓ such that y∈[0,1]↦ f(y) is
continuous. We write c_0=0,c_ℓ+1=1, and 
I_1=[c_0,c_1),I_2=(c_1,c_2),…,I_ℓ+1=(c_ℓ,c_ℓ+1] for the intervals of monotonicity.
For each y∈[0,1], let f_i(y) be f(y)_I_i, see Figure <ref>.


Let 0=a_1<b_1<a_2<b_2<⋯<a_ℓ+1<b_ℓ+1=1, and let
k:[a_1,b_1]∪⋯∪[a_ℓ+1,b_ℓ+1]→[0,1]
be a differentiable map such that
k({a_i,b_i})⊂{0,1} and | k'|>γ>1. 
We define K(x,y) by

    K(x,y)=K_i(y), for  x∈ I_i,

where K_i=(k_[a_i, b_i])^-1, i=1,…, ℓ+1, are the inverse branches of k.

Generalized Toy Model: It is the map defined by

    [                       F  : ([0,1]\{c_1,…,c_ℓ})× [0,1]                          ⟶                [0,1]×[0,1];                                                 (x,y)                          ⟼         (f(y)(x), K(x,y)). ]

We can similarly introduce extra points to make the domain of F become compact.
For each i=1,…,ℓ, introduce points c_i^i and c_i^i+1 and extend F via the formula
F(c^i_i,y)=(f_i(y)(c_i),K_i(y)) and F(c^i+1_i,y)=(f_i+1(y)(c_i),K_i+1(y)), see Figure <ref>.



For  i=1,…,ℓ+1, let I_i be an interval on the construction of the family y↦ f(y). 
We make the following convention.
If f(y) is strictly increasing on I_i, we put

    [                                        F_i  :                        [c_i-1^i,c_i^i]× [0,1]                                             ⟶ ([0,c_1^1]∪(c_1^2,c^2_2]∪⋯∪(c_ℓ^ℓ+1,1])×[0,1];                                                                                       (x,y)                                             ⟼                          (f_i(y)(x), K_i(y)). ]

If f(y) is strictly decreasing on I_i, we put

    [                                        F_i  :                        [c_i-1^i,c_i^i]× [0,1]                                             ⟶ ([0,c_1^1)∪[c_1^2,c^2_2)∪⋯∪[c_ℓ^ℓ+1,1])×[0,1];                                                                                       (x,y)                                             ⟼                          (f_i(y)(x), K_i(y)). ]

For each (x,y)∈Dom(F), there exists
j(x,y)=(j_1j_2⋯ j_m⋯)∈{1,…, ℓ+1}^
such that

    F^m+1(x,y)=F_j_m+1(x_m,y_m), ∀ m≥ 0,

where y_m=(K_j_m∘⋯∘ K_j_1)(y)=:y_j_m⋯ j_1 and
x_m=f_j_m(y_j_m-1⋯ j_1)∘⋯∘ f_j_2(y_j_1)∘ f_j_1(y)(x). 
Consider the alphabet 𝒜={I_1,c_1^1,c_1^2,I_2,…,
c_ℓ^ℓ,c_ℓ^ℓ+1,I_ℓ+1}.

Address of a point: The address of a point (x,y)∈Dom(F) is 
the letter t_F(x,y)∈𝒜 defined by

    t_F(x,y)= {[                    I_i     , if π_1(x,y)∈ I_i;                  c_i^i    , if π_1(x,y)=c^i_i;                c_i^i+1 , if π_1(x,y)=c^i+1_i. ].


Itinerary of a point: The itinerary of a point (x,y)∈Dom(F)
is the sequence T_F(x,y)∈𝒜^{0,1,2,…} defined by 

    T_F(x,y)=(t_F(x,y),t_F(F(x,y)),…,t_F(F^n(x,y)),…).



Critical Line: The critical line of F is the set

    _c(F)={(c_i^i, y),(c_i^i+1,y): y∈[0,1] and i=1,…, ℓ}.
 


Kneading Sequences: The kneading sequences of F are all the itineraries
of elements of _c(F). We denote this set of itineraries by 𝒦_F(_c). 

With these more general definitions, a result similar to Theorem <ref> holds. 


Let F,G be two generalized toy models, and assume that
neither of them has wandering intervals, interval of periodic points nor
weakly attracting periodic points. Then F,G are fiber topologically conjugate 
if and only if they have the same kneading sequences.


The proof of Theorem <ref> follows exactly the same ideas and arguments
as in the proof of Theorem <ref>, but it demands a much heavier notation. Therefore we decided
not to include its proof and leave the details to the interested reader.













§ PROOFS OF THE MAIN RESULTS


We now present the proofs of our main results. Given a set X, let ∂ X denote its boundary.



 §.§ Proof of Theorem <ref>


A similar proof of Theorem <ref> can be found in <cit.>.
Remind that

    (F)={(x,y)∈Dom(F):∃ ℓ≥0 
     such that  F^ℓ(x,y)∈_c(F)}

and that for y∈[0,1] and n≥1 we have

    _n^F(y):={(x,y)∈Dom(F)∩ I(y):
     F^ℓ(x,y)∈_c(F) 
     for some  0≤ℓ≤ n-1}.

Setting

    ^F(y):=⋃_n≥ 1_n^F(y),



we have

    (F)=∐_y∈[0,1]^F(y).



We can similarly define the sets (G),_n^G(y),^G(y).
Suppose that, for all y∈[0,1] and n≥1, there exists a strictly 
increasing bijection H_n(y):_n^F(y)→_n^G(ψ(y)) such that:


  * H_n(y)∘ F=G∘ H_n(y) on _n^F(y)\{(0^±,y)},

  * H_n(y)__n-1^F(y)=H_n-1(y).

Since ψ is a bijection, we have

    (G)=∐_y∈[0,1]^G(ψ(y)).

Therefore, we can define H:(F)⟶(G) by
H(x,y)=H_n(y)(x,y) where n is some (any) integer for which (x,y)∈_n^F(y).
It follows directly that H∘ F=G∘ H on 𝒞(F)\_c(F).
Hence the proof of Theorem <ref> is reduced to 
constructing the maps H_n(y). This is what we will now do.

Let _n^F(y) be the partition of I(y) induced by _n^F(y).
More specifically,

    _n^F(y):={I_i^F(y)⊂ I(y):∂ I_i^F(y)
    ⊂_n^F(y)∪{(-1,y), (1,y)},
    1≤ i≤ k_n^F(y)},

where k_n^F(y):=#_n^F(y) and the increasing order of the index of 
I_i^F(y) is the same as the intervals are placed in I(y). Note that the partition _n^F(y)
includes two intervals, one with left endpoint 0^+ and one with
right endpoint 0^-.

Given a sequence
j=j_1j_2⋯ j_m ∈{-, +}^m, we will use the following notation 

    f_j(y):=f_j_m(y_j_m-1⋯ j_1)∘⋯∘ 
    f_j_3(y_j_2j_1)∘ f_j_2(y_j_1)∘ f_j_1(y),

where y_j_ℓ⋯ j_1=(K_j_ℓ∘⋯∘ K_j_1)(y)
for 1≤ℓ≤ m. 

Given (x,y)∈_n^F(y)\{(0^±,y)}, we will identify the pair (x,y)
with x(y)_j(x,y)^F, where j(x,y)=j_1j_2⋯ j_k ∈{-, +}^k is the minimal sequence such that

    F^k(x(y)_j(x,y)^F)=(f_j(x,y)(y)(x(y)_j(x,y)^F), 
    y_j_k⋯ j_1)=(0^j_k, y_j_k⋯ j_1)

for some 1≤ k≤ n-1.
We will also abuse notation
by considering x(y)_j(x,y)^F either as an ordered pair in order to apply F or as a real number
in order to apply unimodal maps.

For each I_i^F(y)∈_n^F(y) there exists a unique minimal sequence
j(I_i^F(y))=j^i_1j_2^i⋯ j_n^i∈{-,+}^n
such that the function 

    (z,y)∈int(I_i^F(y))↦(f_j(I_i^F(y))(y)(z),y_j^i_n ⋯ j^i_1)
 
is strictly monotone. Hence

    ∂[I_i^F(y)]⊂{(-1,y), (1,y), (0^±,y), 
    x(y)_j^i_1j_2^i⋯ j_k^i^F}

for some 1≤ k≤ n-1. For each n≥1, set

    A_n^F(y):={j_1j_2⋯ j_k∈{-, +}^k:  
    [         ∃ (x,y)∈_n^F(y)\{(0^±,y)} such that; (x,y)=x(y)^F_j_1j_2⋯ j_k for some 1≤ k≤ n-1 ]}.

We similarly define _n^G(ψ(y)) and A_n^G(ψ(y)). 
Now, we will use induction on n to complete the proof of the theorem.

Our underlying assumptions imply that we can define 
H_1(y):_1^F(y)→_1^G(ψ(y)) with
(0^±,y)↦(0^±,ψ(y)).
By induction, suppose that there exists a strictly increasing map
H_n(y):_n^F(y)→_n^G(ψ(y)) such that
H_n(y)∘ F=G∘ H_n(y) on _n^F(y)\{(0^±,y)} for every y∈[0,1]. 
In particular, #_n^F(y)=#_n^G(ψ(y)) and 
A_n^F(y)=A_n^G(ψ(y)) for all y∈[0,1].

Let I_i^F(y)∈_n^F(y). There is a unique sequence 
j=j_1j_2⋯ j_n∈{-,+}^n with 

    ∂[I_i^F(y)]⊂{(±1, y),(0^±, y),x(y)_j_1⋯ j_k^F}

for some 1≤ k≤ n-1. We analyse the possibilities for the interval I_i^F(y).
The first possibility is

    I_i^F(y)=[[x(y)_j_1⋯ j_k^F, x(y)_j_1⋯ j_ℓ^F],y]

for 1≤ k≠ℓ≤ n-1. By the induction hypothesis, we have

    I_i^G(ψ(y))=[[x(ψ(y))_j_1⋯ j_k^G, 
    x(ψ(y))_j_1⋯ j_ℓ^G],y].

Moreover, H_n(y)(x(y)_j_1⋯ j_k^F)=x(ψ(y))_j_1⋯ j_k^G
and H_n(y)(x(y)_j_1⋯ j_ℓ^F)=x(ψ(y))_j_1⋯ j_ℓ^G.
Hence F^n[I_i^F(y)] is equal to the union

    [             [(f_j_n(y_j_n-1⋯ j_1)
    ∘⋯∘ f_j_k+1(y_j_k⋯ j_1)(0^j_k+1),; f_j_n(y_j_n-1⋯ j_1)
    ∘⋯∘ f_j_ℓ+1(y_j_ℓ⋯ j_1)(0^j_ℓ+1))
    , y_j_n⋯ j_1] ]∪Λ(I_i^F(y))

where Λ(I_i^F(y))={F^n-k(0^j_k,y_j_k⋯ j_1),
F^n-ℓ(0^j_ℓ,y_j_ℓ⋯ j_1)}, and G^n[I_i^G(ψ(y))] is equal to the union

    [             [(g_j_n(ψ(y)_j_n-1⋯ j_1)
    ∘⋯∘ g_j_k+1(ψ(y)_j_k⋯ j_1)(0^j_k+1),; g_j_n(ψ(y)_j_n-1⋯ j_1)
    ∘⋯∘ g_j_l+1(ψ(y)_j_ℓ⋯ j_1)(0^j_ℓ+1))
    , y_j_n⋯ j_1] ]∪ Λ(I_i^G(ψ(y)))

where Λ(I_i^G(ψ(y)))={G^n-k(0^j_k,ψ(y)_j_k⋯ j_1),
G^n-ℓ(0^j_ℓ,ψ(y)_j_ℓ⋯ j_1)}.

Since 𝒦_F(_c)=𝒦_G(_c), we have

    F^n[I_i^F(y)]∩_c(F)≠∅  G^n[I_i^G(ψ(y))]∩_c(G)≠∅.

If F^n[I_i^F(y)]∩_c(F)=∅, 
then I_i^F(y)∈_n+1^F(y).
On the other hand, if 
Λ(I_i^F(y))∩_c(F)≠∅
we get I_i^F(y)∈_n+1^F(y) since
F is injective. In any case, we also have
I_i^G(ψ(y))∈_n+1^G(ψ(y)).

Now, if (F^n[I_i^F(y)]\Λ(I_i^F(y)))
∩_c(F)≠∅ then there is a unique point

    x(y)_j_1j_2⋯ j_n^F:=(f_j_1^-1(y)∘⋯∘
    f_j_n^-1(y_j_n-1⋯ j_1)(0^j_n),y)∈ int(I_i^F(y)),

and so (G^n[I_i^G(ψ(y))]\Λ(I_i^G(ψ(y))))
∩_c(G)≠∅, which in turn implies the existence of a unique point

    x(ψ(y))_j_1j_2⋯ j_n^G:=(g_j_1^-1(ψ(y))∘⋯∘
    g_j_n^-1(ψ(y)_j_n-1⋯ j_1)(0^j_n),ψ(y))∈int(I_i^G(ψ(y))).

Therefore we can define 

    H_n+1(y)(x(y)_j_1j_2⋯ j_n^F):=x(ψ(y))_j_1j_2⋯ j_n^G.

The other possibilities I_i^F=[[-1, x(y)^F_j_1j_2⋯ j_k], y],
I_i^F=[[x(y)^F_j_1j_2⋯ j_k, 1], y],
and I_i^F=[[x(y)^F_j_1j_2⋯ j_k, 0^±], y], 1≤ k≤ n-1, are treated similarly.
Finally, defining H_n+1(x,y)=H_n(x,y) for (x,y)∈_n^F(y), we have just constructed
a strictly increasing map

    H_n+1(y):_n+1^F(y)→_n+1^G(ψ(y))

such that:


  * H_n+1(y)∘ F=G∘ H_n+1(y) on _n+1^F(y)\{(0^±,y)}, and

  * H_n+1(y)__n^F(y)=H_n(y). 

This concludes the proof of Theorem <ref>.





 §.§ Proof of Theorem <ref>


First, we shall prove some auxiliary lemmas.


Let y↦ f(y) be the one-parameter family of unimodal maps, as in the definition of toy models.
For every y∈[0,1] and ε>0
there is δ>0 satisfying the following: if y'∈[0,1], x,x'∈[-1,1]
are such that d(y,y')<δ, d(x,x')<δ and there exist
x(y)=f^-1_j(y)(x) and x'(y')=f^-1_j(y')(x'), where j=-  or  +,
then d(x(y),x'(y'))<ε.



The proof is by contradiction.
Suppose there is ε>0 such that for all n≥1 there are
y_n∈[0,1] and x_n^y,x_n^y_n∈[-1,1] with d(y,y_n)<1n 
and d(x_n^y,x_n^y_n)<1n such that x_n(y)=f^-1_j(y)(x_n^y) and
x_n(y_n)=f^-1_j(y_n)(x_n^y_n) exist and d(x_n(y),x_n(y_n))≥ε.
By compactness, we may suppose that lim x_n^y=lim x_n^y_n=x_0,
lim x_n(y)=x_1 and lim x_n(y_n)=x_2.
Note that x_1≠ x_2 and that either x_1,x_2∈[-1,0] or x_1,x_2∈[0,1]. 
In particular f(y)(x_1)≠ f(y)(x_2). But, using that f(y_n) 
converges uniformly to f(y), we have
f(y)(x_1)=lim f(y)(x_n(y))=lim x_n^y=lim x_n^y_n=lim f(y_n)(x_n(y_n))=f(y)(x_2),
a contradiction.


Let ξ_-:([-1,0^-]∪(0^+,1])×[0,1]→[-1,0^-] and 
ξ_+:([-1,0^-)∪[0^+,1])×[0,1]→[0^+,1]
be two maps defined by 

    ξ_j(x,y)= {[        f_j^-1(y)(x) , if   x∈Im(f_j(y));                 0^j        , otherwise. ].

Lemma <ref> has the following consequence.


The maps ξ_j defined above are continuous.



Let (x,y)∈Dom(ξ_j) and ε>0. Assume that 
x∈[-1, f_j(y)(0^j)). Since the family y↦ f(y)
depends continuously on y, there exists δ>0 such that
for every (x',y')∈Dom(ξ_j) satisfying d(x,x')<δ and d(y,y')<δ,
we have x'∈[-1, f_j(y')(0^j)).
From Lemma <ref>, if δ>0 is small enough then

    d(ξ_j(x,y),ξ_j(x',y'))=d(f^-1_j(y)(x),f^-1_j(y')(x'))<ε.

Suppose now that x>f_j(y)(0^j). Again by continuity, there exists
δ>0 such that for all (x',y')∈Dom(ξ_j) satisfying d(x,x')<δ 
and d(y,y')<δ we have x'>f_j(y')(0^j). Therefore,
ξ_j(x,y)=0^j=ξ_j(x',y') and so 

    d(ξ_j(x,y),ξ_j(x',y'))=0.

The case x=f_j(y)(0^j) is treated analogously.



The following lemma is fundamental to prove the 
continuity of the conjugacy in Theorem <ref>.


For every (x,y)∈(F) there exists a continuous curve 
γ^F:[0,1]→(F) of the form γ^F(w)=(γ^F(w),w) 
such that γ^F(y)=(x,y).



Given (x,y)∈(F) there exists n≥1 and a 
sequence (j_1⋯ j_n)∈{-,+}^n
such that (x,y)=(f^-1_j_1(y)∘⋯∘ 
f^-1_j_n(y_j_n-1⋯ j_1)(0^j_n),y).
Consider a sequence of curves defined inductively as follows:


  * γ^F_j_1:[0,1]→Dom(F) is defined by 
γ^F_j_1(w)=(ξ_j_1(0^j_1,w),w).

  * γ^F_j_1j_2:[0,1]→Dom(F) is defined by 
γ^F_j_1j_2(w)=(ξ_j_1(π_1(γ^F_j_2(w_j_1)),w),w).

  * ⋯

  * γ^F_j_1⋯ j_n:[0,1]→Dom(F) is defined by 
γ^F_j_1⋯ j_n(w)=(γ^F_j_1⋯ j_n(w),w),
where γ^F_j_1⋯ j_n(w):=
ξ_j_1(π_1(γ^F_j_2⋯ j_n(w_j_1)),w).

From Corollary <ref>, each γ^F_j_1⋯ j_n:
[0,1]→Dom(F) is a continuous curve. 
By construction, γ^F_j_1⋯ j_n(y)=(x,y).
Now it is quite easy to see that γ^F_j_1⋯ j_n([0,1])⊂(F).
For each w∈[0,1], put 

    ℓ_j_1⋯ j_n(w):=max{
    1≤ k≤ n: f^-1_j_1(w)∘⋯∘ 
    f^-1_j_k(w_j_k-1⋯ j_1)(0^j_k) exists}.

Hence, if ℓ_j_1⋯ j_n(w) exists then

    γ^F_j_1⋯ j_n(w)=(f^-1_j_1(w)∘⋯∘ 
    f^-1_j_ℓ_j_1⋯ j_n(w)(w_j_ℓ_j_1⋯ j_n
    (w)-1⋯ j_1)(0^j_ℓ_j_1⋯ j_n(w)),w).

On the other hand, if ℓ_j_1⋯ j_n(w) does not exist, then

    γ^F_j_1⋯ j_n(w)=(0^j_1,w).

In any case we have γ^F_j_1⋯ j_n([0,1])⊂(F).


Here is a direct consequence of Lemma <ref>:
if (x,y),(z,w)∈(F) are such that ∃ n≥1 and
(j_1⋯ j_n)∈{-,+}^n
with (x,y)=(f^-1_j_1(y)∘⋯∘ 
f^-1_j_n(y_j_n-1⋯ j_1)(0^j_n),y) and 
(z,w)=(f^-1_j_1(w)∘⋯∘ 
f^-1_j_n(w_j_n-1⋯ j_1)(0^j_n),w),
then there is a continuous curve 
γ^F:[0,1]→(F) such that γ^F(y)=(x,y) and γ^F(w)=(z,w).  



Because of Proposition <ref>, we only need to prove the reverse implication.
We will employ similar notation to the one used in the proof of Theorem <ref>.
For y∈[0,1], let H(y):^F(y)→^G(ψ(y)) be the map defined by
H(y)(x,y):=H_n(y)(x,y),
where n is some (any) integer such that (x,y)∈_n^F(y). We will extend 
H(y) to I(y). Take (z,y)∈^F(y)\^F(y)
and suppose that there exist (z_n,y)^j∈^F(y), j=1,2, such that (z_n,y)^1↑ (z,y)
and (z_n,y)^2↓ (z,y). The cases where we have only 
(z_n,y)↑ (z,y) or (z_n,y)↓ (z,y) are similar. Since G has no wandering
intervals, no intervals of periodic points and no weakly attracting
periodic point, ^G(ψ(y)) is dense on I(ψ(y)).
Thus the limits lim H(y)((z_n,y)^1)=:H^1(y)(z,y) and 
lim H(y)((z_n,y)^2)=:H^2(y)(z,y) exist and H^1(y)(z,y)=H^2(y)(z,y). 
Therefore, for (z,y)∈^F(y)\^F(y) we can define

    H(y)(z,y):=lim H(y)(z_n,y),

where (z_n,y)∈^F(y) is some (any) sequence such that (z_n,y)→(z,y).
Since ^F(y) is dense on I(y), we obtain the map H:Dom(F)→Dom(G) 
defined by H(x,y):=H(y)(x,y). Note that H is strictly increasing on each fiber I(y).


Claim: The map H is continuous. 



Let (x,y)∈Dom(F)\{(±1,y),(0^±,y): y∈[0,1]} (the case
(x,y)∈{(±1,y),(0^±,y): y∈[0,1]} can be treated similarly). 
Let B_ε(H(x,y)) be a neighborhood of H(x,y) for some ε>0.
Since ^G(ψ(y))=I(ψ(y)),
there are z_1<x<z_2 such that (z_1,y),(z_2,y)∈(F) and
H(z_i,y)∈ B_ε(H(x,y))∩ I(ψ(y)), i=1,2. Note that 
π_1[H(z_1,y)]<π_1[H(x,y)]<π_1[H(z_2,y)]. 
By Lemma <ref>,  there are 
continuous curves γ^G_j_1⋯ j_ℓ and γ^G_e_1⋯ e_k 
such that γ^G_j_1⋯ j_ℓ(ψ(y))=H(z_1,y) and 
γ^G_e_1⋯ e_k(ψ(y))=H(z_2,y). By continuity, there are
y_1<y<y_2 so that the domain D(H(x,y)) whose boundary is formed by the two curves 
γ^G_j_1⋯ j_ℓ([ψ(y_1),ψ(y_2)]),
γ^G_e_1⋯ e_k([ψ(y_1),ψ(y_2)]) and the two intervals
[(γ^G_j_1⋯ j_ℓ(ψ(y_1)),γ^G_e_1⋯ e_k(ψ(y_1))), ψ(y_1)], 
[(γ^G_j_1⋯ j_ℓ(ψ(y_2)),γ^G_e_1⋯ e_k(ψ(y_2))), ψ(y_2)] 
contains H(x,y) and satisfies D(H(x,y))⊂ B_ε(H(x,y)), see Figure <ref>.
Again by Lemma <ref>, for every sequence 
t_1⋯ t_m∈{-,+}^m and w∈[0,1] we get 
H(γ^F_t_1⋯ t_m(w))=γ^G_t_1⋯ t_m(ψ(w)).
Let D(x,y) be the domain whose boundary is formed by the curves
γ^F_j_1⋯ j_ℓ([y_1,y_2]),
γ^F_e_1⋯ e_k([y_1,y_2]) and the  intervals
[(γ^F_j_1⋯ j_ℓ(y_1),γ^F_e_1⋯ e_k(y_1)), y_1], 
[(γ^F_j_1⋯ j_ℓ(y_2),γ^F_e_1⋯ e_k(y_2)), y_2]. 
Such domain contains a neighborhood of (x,y). 
Since H is monotone on fibers, we have H(D(x,y))=D(H(x,y)). 
This proves the claim.




At the moment, we know that H is injective (by construction) and surjective (by the claim).
It remains to show that H∘ F=G∘ H. From Theorem <ref>,
we already have H∘ F=G∘ H on (F)\_c(F).
Now, take (z,y)∈Dom(F)\(F)
and let (z_n,y)∈^F(y)\{0^±,y} such that lim(z_n,y)=(z,y)
(the case (z,y)∈_c(F) is treated analogously).
Using that F is continuous on Dom(F)\(F) and that
H(z,y)∈^G(ψ(y))\^G(ψ(y)), we conclude that

    (H∘ F)(z,y)=lim (H∘ F)(z_n,y_n) = lim (G∘ H)(z_n,y_n)=(G∘ H)(z,y).

This finishes the proof of the theorem.


We finish this section remarking that the same proof above can be used
to prove a variant of Theorem <ref>, with assumptions only on G: 
Let F,G be two toy models, and assume that G has no wandering intervals,
no interval of periodic points nor weakly attracting periodic points.
If F,G have the same kneading sequences, then they are fiber topologically semiconjugate.







 §.§ Proof of Theorem <ref>

We begin with a lemma whose proof is similar to that of Proposition <ref>.


Let F be a toy model. If F has no wandering domain, then 

    (F)= Dom(F).




By contradiction, assume that there exists an open set V⊂Dom(F) such that 
F^m(V)∩_c(F)=∅ for all m≥0.
Construct, as in the proof of Proposition <ref>, 
a connected set L containing V such that F^ℓ(L)⊂ L for some 0<ℓ≤ m,
F^i(L)∩_c(F)=∅ and F^i(L)∩ F^j(L)=∅
for all i,j=0,…,ℓ-1 with i≠ j.
Therefore, there exists a sequence j_0j_1⋯ j_ℓ-1∈{-,+}^ℓ
such that, for any (a,b)∈ L and n≥0 
it holds F^nℓ(a,b)=(a_nℓ,b_(j_ℓ-1⋯ j_1j_0)^n), where 
b_(j_ℓ-1⋯ j_1j_0)^n:=(K_j_ℓ-1∘⋯∘ K_j_0)^n(b)  
and a_nℓ=f_j_ℓ-1(b_j_ℓ-2⋯ j_0(j_ℓ-1⋯ j_0)^n-1)
∘⋯∘ f_j_0(b_(j_ℓ-1⋯ j_0)^n-1)∘⋯∘
f_j_ℓ-1(b_j_ℓ-2⋯ j_0)∘⋯∘ 
f_j_1(b_j_0)∘ f_j_0(b)(a). Since K_j_ℓ-1∘⋯∘ K_j_0 
is a contraction, there is w∈π_2(L) such that
(K_j_ℓ-1∘⋯∘ K_j_0)(w)=w. For each (x,y)∈ V, we can 
prove that y=w. Therefore V⊂ I(w), 
which contradicts the fact that V is an open set. 


For n≥1, let

    _n^F={(x,y)∈Dom(F): F^ℓ(x,y)∈_c(F) for some 0≤ℓ≤ n-1}.


For each y∈[0,1], let H_n(y):_n^F(y)→_n^G(ψ(y)) 
be the map constructed in the proof of Theorem <ref>. 
Let H_n(y):I(y)→ I(ψ(y)) be the homeomorphism such that:


  * H_n(y)(±1,y)=(±1,ψ(y)),

  * H_n(y)__n^F(y)=H_n(y), and

  * H_n(y) is linear in each connected component of I(y)\_n^F(y).

Now define H_n:Dom(F)→Dom(G) by
H_n(x,y):=H_n(y)(x,y). 


For each n≥1, H_n is continuous.  



We start invoking the curves constructed in the proof of Lemma <ref>: 
for each j_1j_2⋯ j_ℓ∈{-,+}^ℓ with 0≤ℓ≤ n-1, let
γ^F_j_1j_2⋯ j_ℓ:[0,1]→^F_n
be the continuous curve appearing in the construction of _n^F.
Note that _n^F is the union of finitely many such curves:

    ^F_n = 
    ⋃_j_1j_2⋯ j_ℓ ∈ {-,+}^ℓ
    
                      0 ≤ ℓ ≤  n-1γ^F_j_1j_2⋯ j_ℓ([0,1]).

Take (x,y)∈Dom(F)\_n^F, and take 
δ>0 such that  B_δ(x,y)∩_n^F=∅.
Let x_1<x<x_2 such that (x_1,y),(x_2,y)∈_n^F and
_n^F(y)∩[(x_1,x_2),y]=∅ (if (x_1,y) or (x_2,y) does not
exist, we take x_1=-1 or x_2=1 accordingly).
Take curves γ^F_j_1j_2⋯ j_ℓ and 
γ^F_e_1e_2⋯ e_k such that 
γ^F_j_1j_2⋯ j_ℓ(y)=(x_1,y) and  
γ^F_e_1e_2⋯ e_k(y)=(x_2,y), for some 1≤ℓ,k≤ n-1.
For any curve γ^F_t_1t_2⋯ t_i with 1≤ i≤ n-1, we have
γ^F_t_1t_2⋯ t_i(y)∩[(x_1,x_2),y]=∅.
See Figure <ref>.


Choose y_1<y<y_2 and two continuous curves
α_1,α_2:[y_1,y_2]→_n^F such that:


  * α_i(y)=(x_i,y), i=1,2.

  * If D(x,y) is the domain whose boundary is formed by the two curves α_1([y_1,y_2]),
α_2([y_1,y_2]) and the two intervals
[(α_1(y_1),α_2(y_1)),y_1], [(α_1(y_2),α_2(y_2)),y_2],
then D(x,y)∩_n^F=α_1([y_1,y_2])∪α_2([y_1,y_2]). 

Then H_n(D(x,y))=:D(H_n(x,y)) is a domain such that

    D(H_n(x,y))∩_n^G=α_1
    [ψ(y_1),ψ(y_2)]∪α_2[ψ(y_1),ψ(y_2)],

where α_i=H_n(α_i), see Figure <ref>. 



By the definition of H_n, for (z,w)∈ D(x,y) we have
H_n(z,w)=(ξ_x,y(z,w),ψ(w)), where

    ξ_x,y(z,w)=π_1[α_1(ψ(w))]+
    π_1[α_2(ψ(w))]-
    π_1[α_1(ψ(w))]π_1[α_2(w)]-π_1[α_1(w)](z-π_1[α_1(w)]).

In particular, H_n is continuous at (x,y).
Indeed, H_n is continuous on the union D(x,y)∪∂ D(x,y).

Suppose now that (x,y)∈_n^F. Let s∈{-,+}^ℓ
such that (x,y)=(γ^F_s(y),y) for some 0≤ℓ≤ n-1,
and consider (x_j,y_j)→(x,y). For each j≥ 1, there are curves 
γ^F_a_j,γ^F_b_j such that 
(x_j,y_j) belongs to the interval [[γ^F_a_j(y_j),γ^F_b_j(y_j)], y_j]
and [(γ^F_a_j(y_j),γ^F_b_j(y_j)), y_j]∩_n^F(y)=∅.
To prove that  
H_n(x_j,y_j)→H_n(x,y),
it is enough to prove that
any subsequence (x_j_k,y_j_k)
contains a sub-subsequence (x_j_k_ℓ,y_j_k_ℓ) such that
H_n(x_j_k_ℓ,y_j_k_ℓ)→H_n(x,y).
Let (x_j_k,y_j_k) be a subsequence of (x_j,y_j).
There is a sub-subsequence with
(x_j_k_ℓ,y_j_k_ℓ)∈[[γ^F_a(y_j_k_ℓ),γ^F_b(y_j_k_ℓ)], 
y_j_k_ℓ] for fixed a,b. Since 
γ^F_a(y_j_k_ℓ)≤ x_j_k_ℓ≤γ^F_b(y_j_k_ℓ) for every ℓ, 
when passing to the limit ℓ→∞ we obtain γ^F_a(y)≤γ^F_s(y)≤γ^F_b(y).
If γ^F_a(y)=γ^F_b(y)
then H_n(x_j_k_ℓ,y_j_k_ℓ)→H_n(x,y), since

    H_n(γ^F_a(y_j_k_ℓ),y_j_k_ℓ)≤H_n(x_j_k_ℓ,y_j_k_ℓ)
    ≤H_n(γ^F_b(y_j_k_ℓ),y_j_k_ℓ)

and 

    lim_ℓ→∞H_n(γ^F_a(y_j_k_ℓ),y_j_k_ℓ)=H_n(γ^F_a(y),y)
    =H_n(γ^F_s(y),y)
       =H_n(γ^F_b(y),y)=lim_l→∞H_n(γ^F_b(y_j_k_ℓ),y_j_k_ℓ).

Assume now that γ^F_a(y)≠γ^F_b(y). Then either 
γ^F_s(y)=γ^F_a(y) or γ^F_s(y)=γ^F_b(y). 
In either case we can build a domain D(x,y) such that (x_j_k_ℓ,y_j_k_ℓ)∈ D(x,y) 
for all ℓ≥1. Hence H_n(x_j_k_ℓ,y_j_k_ℓ)→H_n(x,y). 
It follows that H_n is continuous.


Our next goal is to prove that {H_n}_n≥1 is a Cauchy sequence.


The sequence {H_n}_n≥1 is Cauchy.  



Let ε>0. By the equicontinuity of the family {ϕ_n^G}, there is
δ>0 such that if |y_1-y_2|<δ then 
d_ℋ(_n^G(y_1),_n^G(y_2))<ε for all n≥1.
By Lemma <ref>,
(G) is dense in Dom(G), so there is N>0 large enough
such that {B_δ(x,y);  (x,y)∈_N^G} covers Dom(G). 
Let (x,y)∈Dom(F) and m,n≥ N. 
Since H_n is surjective, we can take x_1<x<x_2 such that
|H_n(x_1,y)-H_n(x,y)|=
|H_n(x_2,y)-H_n(x,y)|=2ε. Take
(x_1',y'),(x_2',y”)∈_N^F such that 
H_n(x_1,y)∈ B_δ(H_N(x_1',y'))
and H_n(x_2,y)∈ B_δ(H_N(x_2',y”)). 
Again by equicontinuity, there are
(x_1,y),(x_2,y)∈^F_N such that
H_n(x,y) belongs to the interval [[H_N(x_1,y),
H_N(x_2,y)], ψ(y)]
and |H_N(x_1,y) - 
H_N(x_2,y)|<6ε. Since 
H_ℓ__N^F=H_N__N^F 
∀ℓ≥ N,
we obtain that H_n(x,y),H_m(x,y) both belong to the interval
[[H_N(x_1,y),H_N(x_2,y)], ψ(y)]. Hence

    |H_n(x,y) - H_m(x,y)|<6ε,
 and so

    H_n - H_m=sup_(x,y)∈Dom(F)
    |H_n(x,y) - H_m(x,y)|<6ε.

This proves that {H_n}_n≥1 is a Cauchy sequence.


Now we finish the proof of Theorem <ref>. By the previous lemma,
H:Dom(F)→Dom(G) defined by H:=limH_n is 
continuous and surjective, so it remains to prove that H semiconjugates F and G.

Since H agrees with H_n on 
^F_n, we have H∘ F=G∘H on (F)\_c(F).
Take (x,y)∈Dom(F)\(F) and (x_n,y_n)∈(F)
such that lim(x_n,y_n)=(x,y). 



Since F is continuous on Dom(F)\(F) and 
H(x,y)∈Dom(G)\(G),

    (H∘ F)(x,y)=lim(H∘ F)(x_n,y_n)=
    lim (G∘H)(x_n,y_n)=(G∘H)(x,y).

Take now (x,y)∈_c(F), say (x,y)=(0^-,y) (the case (x,y)=(0^+,y) is similar). 
If F(0^-,y)∉_c(F), then for any sequence 
(x_n,y_n)∈(F)\_c(F) with lim(x_n,y_n)=(0^-,y) we have lim F(x_n,y_n)=F(0^-,y). Hence
(H∘ F)(0^-,y)=(G∘H)(0^-,y). Now assume that F(0^-,y)∈_c(F), then


    (H∘ F)(0^-,y)=(0^-,ψ∘ K_-^F(y))=(0^-, K_-^G∘ψ(y))
    =G(0^-,ψ(y))=(G∘H)(0^-,y),

where in the third passage we used that the kneading sequences 
𝒦_F(_c) and 𝒦_G(_c) are the same.
Therefore H∘ F=G∘H on Dom(F),
which concludes the proof.











 §.§ Proof of Theorem <ref>


Let f:I→ I be a C^3 interval map and let x∈ I 
such that Df(x)≠0.

Schwarzian Derivative: The Schwarzian derivative
Sf(x) of f at x is

    Sf(x):= D^3f(x)Df(x)-32(D^2f(x)Df(x))^2.


We say that f has negative Schwarzian derivative if Sf(x)<0
for all x∈ I except, possibly, the turning points.
At these points we define Sf(x)=-∞.
It is easily seen from the definition that if f,g:I→ I are C^3 and
x∈ I satisfies Df(x)≠0 and Dg(f(x))≠0 then

    S(g∘ f)(x)=Sg(f(x))·(Df(x))^2 + Sf(x).
 
This implies the following: if f,g have negative Schwarzian derivative,
then g∘ f also has negative Schwarzian derivative.


Let I=[a,b] and f:I→ I a C^3 map with negative Schwarzian derivative. 
If Df(x)≠0 for all x∈ I, then

    |Df(x)|>min{|Df(a)|, |Df(b)|},  ∀ x∈(a,b).




See <cit.>.


We call c∈ I a critical point of f if Df(c)=0. The following theorem, due to Singer,
is a consequence of the Minimum Principle.

[Singer]
If f:I→ I is a C^3 map with negative Schwarzian
derivative then the immediate basin of any 
attracting periodic orbit contains either a critical 
point of f or a boundary point of I.



See <cit.>.


Let us restate Theorem <ref>,
which is a version of Singer's theorem for the toy models.


Let F(x,y)=(f(y)(x), K^F_ sign(x)(y)) be a toy model.
If each interval map f(y) has negative Schwarzian 
derivative, then the closure of the immediate 
basin of any strongly attracting periodic orbit contains either a 
point of the critical line or a point of 
Λ:={-1,1}×[0,1].


Before proving this theorem, let us introduce some notation. Write the derivative 
of F(x,y)=(f(y)(x), K_sign(x)(y)) by

    DF=(
    [ f_x f_y;   0 K_y;     ]).

For (x,y)∈Dom(F), let j_(x,y)=(j_1⋯ j_m⋯)∈{-,+}^
such that

    F^m(x,y)=(f_j_m(y_j_m-1⋯ j_1)∘⋯∘
    f_j_2(y_j_1)∘ f_j_1(y)(x),y_j_m⋯ j_1),  ∀ m≥0.

The derivative of F^m at (x,y) is

    DF^m(x,y)=(
    [ A^m(x,y) B^m(x,y);        0 D^m(x,y);          ]),

where

    [                                                                 A^m(x,y)                                                                        =                                                  ∏_ℓ=0^m-1f_x(F^ℓ(x,y)),;                                                                 D^m(x,y)                                                                        =                                                  ∏_ℓ=0^m-1K_y(F^ℓ(x,y)),;                                                                 B^m(x,y)                                                                        = ∑_j=0^m-1(f_y(F^j(x,y))
    ∏_ℓ=0^j-1K_y(F^ℓ(x,y))∏_ℓ=j+1^m-1f_x(F^ℓ(x,y))). ]




Let (p,q)∈Dom(F) be a strongly attracting periodic point of period m,
and let B be the closure of its immediate basin. 
By contradiction, we assume that B does not contain points of 
Λ neither points of the critical line.
Let B_0 be the interior of the connected component of B that contains (p,q). Then
F^m(B_0)⊂ B_0. Let [(a,b), q] be the connected
component of B_0∩ I(q) containing (p,q), see Figure <ref>.
 

Since F^m(p,q)=(p,q) and F preserves fibers, 
F^m([(a,b), q])⊂([(a,b), q]).
Let j=(j_1⋯ j_m)∈{-,+}^m be the sequence that gives the iterations of F up to order m inside
[(a,b),q], i.e. F^m(x,q)=(g(x), q ) for x∈(a,b), where
g=f_j_m(q_j_m-1⋯ j_1)∘⋯∘ f_j_2(q_j_1)∘ f_j_1(q).
By assumption, we also have A^m(x,q)≠ 0.

Claim: g({a,b})⊂{a,b}. 


Suppose that g(a)∈(a,b) (the case g(b)∈(a,b) is treated analogously).
Since F^m is continuous at (a,q), there is a neighborhood V∋(a,q)
such that F^m(V)⊂ B_0. This contradicts the assumption that B_0 is the
connected component of the immediate basin containing (p,q). The claim is proved.


We can assume that A^m(x,q)>0 for all x∈(a,b) 
and that g(α)=α for α∈{a,b}
(if this is not the case, take A^2m(x,q) instead of A^m(x,q)).
Since (a,q),(b,q) are not attracting periodic points, we have
A^m(α, q)≥ 1 for α∈{a,b}. Since g is the composition of
maps with negative Schwarzian derivative, it also has negative Schwarzian 
derivative. By the Minimum Principle, A^m(x,q)>1 
for all (x,q)∈[(a,b),q], which contradicts the equality F^m([(a,b), q])=[(a,b), q].


We finish this paper with a question. One of the assumptions for Theorem <ref>
is the non-existence of wandering intervals. For unimodal maps,
negative Schwarzian derivative guarantees the absence of wandering intervals.
This is a famous theorem of Guckenheimer.

[Guckenheimer]
Let f:I→ I be a C^3 unimodal map with
negative Schwarzian derivative. If f”(c)≠0 at the unique critical point c of f,
then f has no wandering intervals.



See <cit.>.


We do not know if such implication also holds for toy models, so we ask the following.

Question: Let F(x,y)=(f(y)(x), K^F_ sign(x)(y)) be a toy model.
If each unimodal map f(y) has negative Schwarzian 
derivative, can F have wandering intervals?






















alpha

