# MSc thesis- A quantum pseudo-integrable Hamiltonian impact system

Omer Yaniv  Department of Computer Science and Applied Mathematics,Weizmann
Institute of Science, Rehovot 7610001, Israel

## Abstract

A quantization of a toy model of a pseudointegrable Hamiltonian impact system
is introduced, including EBK quantization conditions, a verification of Weyl’s
law, the study of their wavefunctions and a study of their energy levels
properties. It is demonstrated that the energy levels statistics are similar
to those of pseudointegrable billiards. Yet, here, the density of
wavefunctions which concentrate on projections of classical level sets to the
configuration space does not disappear at large energies, suggesting that
there is no equidistribution in the configuration space in the large energy
limit; this is shown analytically for some limit symmetric cases and is
demonstrated numerically for some nonsymmetric cases.

##  1 Introduction

Hamiltonian systems are a mathematical branch of the theory of dynamical
systems, which investigates the time-evolution of a system’s state, attempting
to provide information on the evolution of all possible initial states of the
system. When such a system evolves continuously, it is called a continuous
dynamical system, and in the smooth case it is described by a set of
differential equations.

###  1.1 Classical Hamiltonian dynamical systems

Hamiltonian systems arise in various fields of Physics: for example, they
describe the dynamics of a body moving under conservative forces.

A Hamiltonian system on a symplectic manifold [2] [18] $(M,\omega)$, where
$\omega$ is a closed non-degenerate 2 form, e.g
$\omega=\sum_{i=1}^{n}dq_{i}\wedge dp_{i}$ for set of canonical coordinates
$(p,q)$, is defined by a smooth Hamiltonian function $H\in
C^{\infty}(M,\mathbb{R})$, and $X_{H}$ which is the associated vector field to
H:

$dH(Y)=\omega(X_{H},Y)$ (1)

The dynamics of any observable $f$ obeys the equation of motion:

$\frac{d}{dt}f=\\{f,H\\}$ (2)

where $\\{f,H\\}$ are called Poisson brackets and are defined as:
$\omega(X_{f},X_{H})$. In canonical coordinates $(q,p)$, we can write
$X_{H}=\sum_{i=1}^{n}(\frac{\partial H}{\partial
p_{i}}\frac{\partial}{\partial q_{i}}-\frac{\partial H}{\partial
q_{i}}\frac{\partial}{\partial p_{i}})$.
The Hamiltonian flow $F^{t}:M\rightarrow M$ is generated by $X_{H}$ as

$\frac{d}{dt}F^{t}_{t=0}=X_{H}$ (3)

$F^{t}:M\rightarrow M$ is symplectomorphism:

$F^{t*}\omega=\omega$ (4)

Thus,

$\frac{F^{t*}\omega^{n}}{n!}=\frac{\omega^{n}}{n!}$ (5)

where $\omega^{n}$ is the phase space volume differential form
e.g, given a region $R\in M$

$\int_{R}\frac{\omega^{n}}{n!}=\int_{R}dq_{1}\wedge...\wedge dq_{n}\wedge
dp_{1}\wedge...\wedge dp_{n}=Vol(R)$ (6)

Thus, eq. 5 implies that flow $F^{t}$ is volume preserving.

We concentrate here on Hamiltonian systems of the mechanical form, where the
Hamiltonian function is the sum of a diagonal kinetic energy term and a
potential energy defined on $\mathbb{R}^{n}$, and the natural canonical
variables are the momenta, $p_{i}$ and the configuration coordinates
$q_{i},i=1,...n$:

$H(q,p)=\frac{1}{2}\sum_{i=1}^{n}p_{i}^{2}+V(q),\qquad(q,p)\in\mathbb{R}^{n}\times\mathbb{R}^{n}$

These are called $n$ degrees of freedom mechanical Hamiltonian systems. The
goal of the analysis is to determine the evolution of all initial conditions
in the phase space $(q,p)$. In these coordinates the equations of motion are:

$\displaystyle\frac{d}{dt}p_{i}$ $\displaystyle=-\frac{\partial H}{\partial
q_{i}}$ (7) $\displaystyle\frac{d}{dt}q_{i}$ $\displaystyle=\frac{\partial
H}{\partial p_{i}}$

The simplest type of $n$ degrees of freedom Hamiltonian systems are integrable
systems. Such systems exhibit at least $n$ independent smooth constants of
motions which are pairwise in involution (their pair-wise Poisson bracket
vanishes). A full description of the dynamics in such systems is provided by
the Arnold- Liouville theorem [2], which tells us that for almost all values
of $(q,p)$ there exists a transformation
$(q_{i},p_{i})\rightarrow(\theta_{i},I_{i})$ such that $H(q,p)=H(I)$ and the
motion occurs along connected components of level sets of $I$. Each such
component is an $n$ dimensional torus:

$\displaystyle{\dot{I}}_{i}=-{\frac{\partial H}{\partial\theta_{i}}}=0$ (8)
$\displaystyle{\dot{\theta}_{i}}=-{\frac{\partial H}{\partial
I_{i}}}=\omega_{i}$

Moreover, in open neighborhoods at which such transformation exists, it is
smooth.

On the other hand, Hamiltonian systems can also be chaotic, which is an
extreme case of a disordered motion. The definition of chaotic systems is
community dependent[11] [23][24] [21].

In ergodic theory [22], chaos is defined by the global properties of a measure
preserving map on a probability space $(X,\mathcal{B},\mu,T)$ , which in our
case is the symplectomorphism associated with the Hamiltonian flow.
In order to define chaos in such systems we first need to define ergodicity: T
is said to be ergodic if the only T-invariant sets are of measure 0 or of
measure 1.
An ergodic map T is said to have the strong mixing property if for all
$A,B\in\mathcal{B}$, $\lim_{k\rightarrow\infty}\mu(A\cap
T^{-k}(B))=\mu(A)\mu(B)$.

An ergodic map which is strongly mixing is usually said to be chaotic within
the notion of ergodic theory (yet, this definition depends on the measure
$\mu$, and, notably, choosing a "natural invariant measure" is a delicate
issue in ergodic theory. For Hamiltonian systems the natural measure is the
Liouville measure: Lebesgue measure restricted to the energy surface [22]).

A remarkable property of ergodic systems, which is highly important in
physics, relates to the properties of an observable f, defined on the system
$(X,\mathcal{B},\mu,T)$ :

if $T$ is ergodic and $f\circ T\equiv f$, then f is constant almost
everywhere.

From this we conclude the general property that for any measurable function
$g:X\rightarrow\mathbb{R}$ the time average equals the space average:

$\displaystyle\int_{X}gd\mu=\frac{1}{T}\lim_{T\rightarrow\infty}\int_{0}^{\infty}gdt$
(9)

The mathematical study of ergodicity in Hamiltonian systems focuses mainly on
the dynamics of billiards: systems of free particles which reflect elastically
from the billiard table $D$. Formally we take $V(q)\equiv 0$ in the interior
of the billiard table and on its boundary the motion is determined by elastic
reflections:
$(q,p_{\perp})\rightarrow(q,-p_{\perp}),(q,p_{\parallel})\rightarrow(q,p_{\parallel})$
[10].

In this work, we will look at Hamiltonian impact systems (HIS), which is an
extension of the class of billiards: the dynamics is determined by a smooth
Hamiltonian in the table interior and by elastic reflections:
$(q,p_{\perp})\rightarrow(q,-p_{\perp}),(q,p_{\parallel})\rightarrow(q,p_{\parallel})$
on the table’s boundry.

###  1.2 Pseudo integrable billiards

Pseudo integrable billiards arise in the study of plane polygonal rational
billiards (polygonal tables with all corners being rational fractions of
$\pi$).

The path of a particle in a billiard is independent of the energy. For
polygonal billiards, it can be considered as lying in a 3 dimensional space
whose coordinates are the position $q=(q_{1},q_{2})$ and $\theta$, the path
direction. This path lies in a sequence of replicas of the enclosure situated
at different values of $\theta$. In the case of plane polygonal rational
billiards this sequence is finite(the number of possible directions in each
trajectory). Since reflection at an edge is equivalent for continuing the path
into a reflection of the enclosure and the sequence of directions is finite,
the trajectory lies on a 2-dimensional compact surface. Such surfaces are two-
dimensional surfaces of genus $g\geq 1$ [20, 15]. Pseudointegrable dynamics,
correspond to systems with intermediate complexity: they are not ergodic nor
quasi-periodic, there can be level sets that include several ergodic
components as well as bands of periodic orbits.

###  1.3 Quantum dynamical systems and correspondence principle

In the early 20th century, it was clear that physical properties of some
systems cannot be predicted by classical physics. A new theory, quantum
mechanics, was found to be useful in their prediction. According to quantum
mechanics the physical properties of an n-degrees of freedom system are not
defined by smooth function on $\mathbb{R}^{2n}$, but rather is defined by a
Hermitian operator in $Op(\mathcal{S}(\mathbb{R}^{n}))$ where:
$S(\mathbb{R}^{n})=\\{\phi\in
C^{\infty}|\sup_{\mathbb{R}^{n}}|x^{\alpha}\partial^{\beta}\phi|<\infty$ for
all multiindices $\alpha,\beta\\}$.

Given an operator $\hat{F}=Q(F(q_{i},p_{i}))$ with $F\in
C^{\infty}(\mathbb{R}^{2n})$, its time evolution, the operator $\hat{F}$ at
time $t>0$ is:
$\hat{F}_{t}=e^{i\frac{t\hat{H}}{\hbar}}\hat{F}e^{-i\frac{t\hat{H}}{\hbar}}$,
where $\hat{H}=Q(H(q_{i},p_{i}))$ is the operator associated with the
Hamiltonian. Then, the time evolution equation, in analogy to the classical
equation, becomes:

$\displaystyle\dot{\hat{{F}_{t}}}=-i\frac{[\hat{H},\hat{F}_{t}]}{\hbar}$ (10)

Another equivalent form to express the time evolution of the system is via the
time-dependent Schrodinger equation, that considers evolving in time the wave
function $\Psi(q,t)$ rather than the operator:

$\displaystyle i\hbar\frac{\partial}{\partial t}\Psi(q,t)=\hat{H}\Psi(q,t)$
(11)

The time-independent Schrodinger equation is an eigenvalue problem, whose
solutions are called the spectrum of the quantum system:

$\displaystyle\hat{H}\Psi(q)=E\Psi(q)$ (12)

It is believed, that the classical dynamics can be derived as a limit of the
quantum dynamics in the limit of high energies, $\lim\hbar\rightarrow 0$. This
assumption is called the correspondence principle.

In order to relate quantum and classical mechanics we are interested in the
class of transformations $Q:C^{\infty}(\mathbb{R}^{2n})\rightarrow
Op(\mathcal{S}(\mathbb{R}^{n}))$, which are called quantization of classical
phase space [1].
We require Q to satisfy to following conditions:

  * •

Q is linear

  * •

Q(1) = I, where 1 is the constant function 1, and I the identity operator

  * •

for any function $\Phi:R\rightarrow R$ for which $Q(\Phi\circ f)$ and
$\Phi(Q(f))$ are well-defined, $Q(\Phi\circ f)=\Phi(Q(f))$

  * •

The operators $Q(p_{i})$ and $Q(q_{i})$ corresponding to the coordinate
functions $F(p,q)=p_{i}$ , $G(p,q)=q_{i}$ for (i = 1,…,n) are given by
$Q(G)\Psi=q_{i}\Psi$ and
$Q(F)\Psi=-i\hbar\frac{\partial{\Psi}}{\partial{q_{i}}}$

To obtain a correspondence between classical(as described in eq. 2) and
quantum dynamics(as described in eq. 10) we would want to find Q such that:
$Q(\\{f,g\\})=\frac{[Q(f),Q(g)]}{i\hbar}$
However, a quantization Q that satisfies this condition for all f, g $\in
C^{\infty}(\mathbb{R}^{2n})$ altogether with conditions 1 to 4 , mentioned
above, doesn’t exist[1].

The common quantization scheme which is used is Weyl’s quatization[12]:
For a function $a(q,p)\in C^{\infty}(\mathbb{R}^{2n})$ in classical phase
space, $a^{w}$ the operator accepted by Weyl quantization is defined by:

$\displaystyle
a^{w}\Psi(x)=\frac{1}{\hbar^{n}}\int_{\mathbb{R^{n}}}\int_{\mathbb{R^{n}}}e^{\frac{i<x-y,p>}{\hbar}}a(\frac{x+y}{2},p)\Psi(y)dydp$
(13)

Weyl’s quantization satisfies the following condition:

$\displaystyle Q(\\{f,g\\})=\frac{[Q(f),Q(g)]}{i\hbar}+O(\hbar^{2})$ (14)

Thus, in the limit $\hbar\rightarrow 0$ (so called semiclassical limit) we get
a correspondence between classical and quantum dynamics, described by Egorov’s
Theorem [12]:

$\displaystyle\lVert\hat{F}_{t}-Q(F\circ\Phi_{t}(q,p))\rVert=O(\hbar)\text{
where }Q(F)=\hat{F}_{0}$ (15)

Another important theorem regarding classic-quantum correspondence is Weyl’s
theorem[12]: For each $a<b$, we can define the operator $\mathbbm{1}$:

$\displaystyle\mathbbm{1}(\Psi_{E_{j}})=\begin{cases}\Psi_{E_{j}}\phantom{-}\text{
if}:a<{E_{j}}<b\\\ \phantom{-}0\phantom{-}\phantom{-}\text{else}\end{cases}$
(16)

Weyl’s theorem states that:

$\displaystyle N[E_{j}:a\leq E_{j}\leq
b]=\sum_{j}<\mathbbm{1}\Psi_{E_{j}},\Psi_{E_{j}}>=\frac{1}{\hbar^{n}}Vol(a\leq
H\leq b)+o(1)\text{ as }\hbar\rightarrow 0$ (17)

where $E_{j}$ and $\Psi_{E_{j}}$ are the eigenvalues and eigenvectors
(correspondingly) of $\hat{H}=Q(H(q_{i},p_{i}))$ and $H(q_{i},p_{i})$ is the
classical Hamiltonian.

Semmiclassical approximations can also be used within the framework of
Schrodinger equation. An important approximation for one degree of freedom
systems is WKB aproximation [16] which is an approximation for time
independent Schrodinger equation:
In the limit $\hbar\rightarrow 0$, $\Psi(q)$, an eigenfunction of
$\hat{H}=Q(H(q_{i},p_{i}))$ can be approximated by:

$\displaystyle\displaystyle\Psi(x)\approx
C_{0}{\frac{e^{\theta+i\int\hbar^{-1}{\sqrt{2m\left(E-V(x)\right)}}\,dx}}{\hbar^{-1/2}{\sqrt[4]{2m\left(E-V(x)\right)}}}}$
(18)

From this approximation we can derive EBK quantization conditions, that are
used to find the spectrum of integrable systems:

$\displaystyle I(E)=\hbar\left(n+\frac{\mu}{4}+\frac{b}{2}\right)$ (19)

where I is the classical action of a periodic orbit, $\mu$ is the number of
classical turning points along a period and $b$ is the number of impacts from
the table’s boundries along a period [9](chapter 2).

###  1.4 Quantum chaos and ergodicity

Quantum chaos studies how classical dynamics (integrable and non-integrable)
are reflected in the properties (e.g. eigenvalues and eigenfunctions) of the
correspondent quantum system. It is accepted that in integrable systems, the
distribution of the level spacing is provided by the Poisson distribution
$e^{-s}$ [6], while that in chaotic systems (hereafter, meaning mixing system
on energy surfaces, studied by simulating chaotic billiards) they distribute
as eigenvalues of random matrix ensembles (GOE) [8]. When a system has a mixed
phase space, which is the common behavior of smooth Hamiltonian systems, it is
found that a Berry-Robink distribution, a convex hall of the Poisson and the
GOE distributions, describes the level spacing [5, 19]. This distribution
reflects the existence of eigenfunctions supported on the islands of stability
and of eigenfunctions supported on the chaotic components of the classical
phase-space [3].

In quantum pseudo integrbale billiards the level spacing appears to have
intermediate statistics: the nearest-neighbor distribution displays repulsion
at small distances and an exponential decay at large distances [7].

The quantum ergodicity of a system describes the wavefunction properties of
classically ergodic systems, namely that almost all of them are
equidistributed. An important result in quantum ergodicity is the
equidistribution of eigenfunctions:
Let $\\{\Psi_{n}\\}_{n=1}^{\infty}$ be an orthonormal basis of eigenfunctions
of the Laplacian operator on a compact domain D. Provided the billiard flow is
ergodic in phase space, there is a density-one sequence $n_{j}\in\mathbb{N}$
such that for any $A\subset D$:
$\lim_{j\to\infty}\oint_{A}|\Psi_{nj}|^{2}(s)ds=\frac{area(A)}{area(D)}$ [26].

This result was generalized for polygonial billiards with rational angles as
it is known that the motion of a particles in such billiards is dense in
configuration space for almost all directions[17].

##  2 Main results

We investigate eigenvalues statistics and eigenfunctions properties of a class
of systems that belongs to the recently discovered family of classical
pseudointegrable Hamiltonian systems with impacts. Such systems combine motion
under a smooth potential field with continuous symmetries and reflections from
a corresponding family of billiards that keeps the continuous symmetries only
locally and not globally. For example, trajectories of a separable Hamiltonian

$H=H_{1}+H_{2},\ H_{i}(q_{i},p_{i})=\dfrac{p_{i}^{2}}{2m}+V_{i}(q_{i}),\
i=1,2$ (20)

in a right-angled polygonal billiard with at least one concave corner are
pseudointegrable [4, 13].

Here, we study the quantum step oscillators: we take $V_{i}$ to be confining
potentials which are even smooth functions with a single minimum at the origin
and are monotone elsewhere, and take the right angled polygon to be
$\mathbb{R}^{2}\setminus S_{q^{wall}}$, where

$S_{q^{wall}}=\\{(q_{1},q_{2})|\>q_{1}<q_{1}^{wall}\leq 0\text{ and
}q_{2}<q_{2}^{wall}\leq 0\\}.$ (21)

The trajectories are confined by the potential and reflect from the step
$S_{q^{wall}}$ [4], see Figure 1a. Since the step boundaries are parallel to
the axes, the vertical and horizontal momenta are conserved at reflections, so
the motion occurs along the level sets $H_{i}(q_{i},p_{i})=E_{i},\ i=1,2$.
Passing to the action angel coordinates of the smooth separable system,
provided $E_{i}>V_{i}(q_{i}^{wall}),\ i=1,2$, the motion on each level set is
conjugated to the directed motion on the flat cross-shaped surface, see Figure
1b. The direction of motion on this surface is given by
$\omega_{2}(E_{2})/\omega_{1}(E_{1})$ and the cross shaped concave corners are
at $\\{\pm\theta_{1}^{wall}(E_{1}),\pm\theta_{2}^{wall}(E_{2})\\}$, where
$\omega_{i}(E_{i})$ denotes the frequency of the smooth periodic motion under
$H_{i}$ and $\theta_{i}^{wall}(E_{i})$ denotes the angle of an impacting
trajectory (with the convention that $\theta_{i}=0$ at the maximum of
$q_{i}$). So, the direction of motion and the surface dimensions depend
continuously on $(E_{1},E_{2})$. For the case of harmonic oscillators, i.e.
when $V_{i}(q_{i})=\frac{1}{2}\omega_{i}q_{i}^{2}$, the frequencies are fixed
at $\omega_{i}$ and the values of $\theta_{i}^{wall}(E_{i})$ can be explicitly
computed. Equivalently, by folding the surface, the motion on such level sets
is conjugated to the directed billiard motion on an L-shaped billiard, see
Figure (1)c. Thus, this system is pseudointegrable [4]. In general, the
dynamics on such surfaces has non-trivial ergodic properties. It was proven
that if $q_{i}^{wall}<0$ for $i=1,2$, the motion is typically uniquely
ergodic, and, for the case of resonant harmonic oscillators, there are level
sets with co-existing periodic ribbons and dense orbits on some parts of the
cross-shaped surface [13].

(a)

(b)

(c)

Figure 1: A trajectory of a separable Hamiltonian reflecting from a step. (a)
Projection to the configuration space. (b) The corresponding directed motion
on the cross-shaped surface in the angles space. (c) Folding the surface to
the lower left quadrant leads to the corresponding billiard motion on an
L-shaped billiard. Here, Eq. (20) are integrated with elastic reflections from
the step of Eq. (21), with
$V_{i}(q_{i})=\frac{1}{2}\omega_{i}q_{i}^{2},\omega_{1}=1,\omega_{2}=\sqrt{2},q_{1}^{wall}=q_{2}^{wall}=-1,E_{1}=5.625,E_{2}=5.50$.

In this work we described the quantum-classical correspondence in pseudo-
integrable HIS. More specifically:

  * •

We describe classical and quantum dynamics of the pseudo-integrable step
system for the case of a step in the origin and
resonant($\omega_{1}=1,\omega_{2}=\frac{m}{n}$) harmonic potentials (all
regular trajectories are periodic). In particular, we prove the existence of
two families of periodic orbits co-exisisting in the same level set when n is
even, and the exsitence of one family of periodic orbits when n is odd. for
m=1 we find the number of impacts and turning points along a period in each
family and predict the allowed energies using EBK quantization conditions.

  * •

We describe the quantum mechanical properties(eigenfunctions structure and
eigenvalues) for the more general pseudo-integrable systems (i.e system which
are not completely periodic):

    1. 1.

We find level-spacing distribution of the pseudo-integrable step system to be
similar to the level-spacing distribution of the pseudo-integrable
billiards(i.e semi-Poisson distribution)

    2. 2.

For a step in the origin and anharmonic potential(for that case we give
analytical results) we find that at least a fraction of $\frac{1}{3}$ of the
eigenfunctions concentrate on the classical level sets.

    3. 3.

For a step not in the origin and harmonic potential we present numerical
evidence that it is also likely to have positive fraction of the
eigenfunctions which concentrate on the classical level sets.

For achieving that we used both analytic and numerical methods. The numerical
method used is finite differences method via MATLAB for solving the time
independent Schrodinger equation. We used the grid $[x,y]=[-15,15]X[-15,15]$
where $\delta_{x}=0.05$ and $\delta_{y}=0.05$, we set the potential to be
$10^{28}$ at the step.

These main results have been submitted for publication and are currently under
review [25]

##  3 results

###  3.1 Quantization of Periodic orbits

As we are interested in quantization, and, in particular, in studying the role
of superscars(concentration of the wave functions along the classical families
of periodic orbits) in the system, we look first for families of periodic
orbits. Given a family of periodic orbits on a given level set
$(E_{1},E_{2}=E-E_{1})$, with $\mu=(\mu_{1},\mu_{2})$ turning points
($\mu_{1}$ in the horizontal direction and $\mu_{2}$ in the vertical one), and
$b=(b_{1},b_{2})$ impacts ($b_{1}$ with the right side of the step and $b_{2}$
with the upper part of the step), and an action $I(E;\mu,b)$, we can quantize
it by using the EBK quantization conditions [9, 16]:

$\displaystyle
I(E;\mu,b)=\hbar(n+\frac{\mu_{1}+\mu_{2}}{4}+\frac{b_{1}+b_{2}}{2}).$ (22)

Moreover, denoting by $I_{i}(E_{i})$ the action of the smooth $H_{i}$ system
and by $I_{i}^{wall}(E_{i})=\int_{q_{i}\geq
q_{i}^{wall}}p_{i}(q_{i};E_{i})dq_{i}=I_{i}\frac{2\theta_{i}^{wall}}{2\pi}$
the action of the impact $H_{i}$ system, we obtain:

$\displaystyle
I(E_{1},E_{2};\mu,b)=\sum_{i=1}^{2}b_{i}I_{i}^{wall}+(\frac{\mu_{i}-b_{i}}{2})I_{i},$
(23)

namely, given $\mu$, $b,I_{i}(E_{i})$ and $\theta_{i}^{wall}(E_{i})$, we
expect that the EBK quantization rule will predict the energy levels. Yet, in
general, it is non-trivial to find $\mu$ and $b$ (see e.g. section 7 in [13])
nor to invert $I(E_{1},E_{2};\mu,b)$ on the given family of periodic orbits.

We consider first some simple limit cases in which periodic motion can be
easily identified. When the step is at the origin
($S_{0}=S_{q_{1}^{wall}=q_{2}^{wall}=0}$), the corner angles are fixed at
$\theta_{i}^{wall}(E_{i})|_{q_{1}^{wall}=q_{2}^{wall}=0}=\frac{\pi}{2}$, so
the dimensions of the cross-shaped surface are independent of the energy. When
the potentials are harmonic, the direction of motion,
$\frac{\omega_{2}}{\omega_{1}}$ is independent of the energy as well and
$I_{i}=\frac{E_{i}}{\omega_{i}}$. Thus, by choosing resonant harmonic
potentials and a step at the origin, we conclude that for all partial energies
the motion is periodic and of the same type and that
$I_{i}^{wall}=\frac{I_{i}}{2}$. In particular, setting:
$\omega_{1}=1,\omega_{2}=\frac{n}{m}$ (with $gcd(n,m)=1$), it can be shown
that there are exactly 2 options for dynamics:

######  Theorem 3.1.

For harmonic oscillator with a step in the origin with $\omega_{1}=1$ and
$\omega_{2}=\frac{m}{n}$ with odd m and n, there is one family of periodic
orbits and the quantization condition is:
$E_{(k)}=\frac{2k}{3n}+\frac{1+\frac{m}{n}}{2}+\frac{n+m}{3n}=\frac{2k}{3n}+\frac{5(m+n)}{6n}$

######  Theorem 3.2.

For harmonic oscillator with a step in the origin with $\omega_{1}=1$ and
$\omega_{2}=\frac{m}{n}$ where n is even and m is odd, there are two families
of periodic orbits where the first family satisfies: $\mu^{I}_{1}=2n$,
$\mu^{I}_{2}=2m$ and the number of impacts $b^{I}_{1}$ and $b^{I}_{2}$ can be
calculated inductively (see appendix). The second family satisfies:
$\mu^{II}_{1}=n$, $\mu^{II}_{2}=m$ and the number of impacts $b^{II}_{1}$ and
$b^{II}_{2}$ can be calculated inductively(see appendix). For the case of m=1
we can directly calculate that the number of impacts is $b^{I}_{1}=n$,
$b^{I}_{2}=0$ and $b^{II}_{1}=0$, $b^{II}_{2}=1$

Note that on a given resonant level set
$m\omega_{1}(E_{1})=n\omega_{2}(E_{2})$, by rescaling time and energy:
$\hat{t}=\omega_{1}t,\hat{H}=\frac{H}{\omega_{1}}$, we can set
$\hat{\omega}_{1}=1$ and $\hat{\omega}_{2}=\frac{m}{n}$. Clearly $b_{i}$ and
$\mu_{i}$ are unchanged by the rescaling of time, so we study their dependence
on $\frac{m}{n}$ for the rescaled time case. Equivalently, the direction of
the flow on the crossed surface depends only on the ratio of the frequencies.
Finally by replacing the roles of $\theta_{1}$ and $\theta_{2}$ Theorem 3.2
can be applied to the case of odd n and even m.

In figure 8, we validate the above results(Theorems 3.1 and 3.2). Notice that
for even $m$ there are infinite number of energy levels at which
$E^{I}_{k_{1}}=E^{II}_{k_{2}}$ (marked with green lines), and in particular,
for $m=2,n=1$ and $m=2,n=5$, $E^{I}_{k_{1}}=E^{II}_{2k_{1}}$ (as shown in Fig.
8). Since the system here is symmetric, all these energy levels are
degenerate, and, as shown in 8b and 8c, the common energy levels for the two
families have higher degeneracy.

(a)

(b)

(c)

Figure 2: Energy levels for resonant harmonic oscillator with a step at the
origin: numerical and expected (EBK) values. (a) Odd $m$ ($\omega_{1}=1,\
\omega_{2}=1$): the expected values for the single family of the periodic
orbits denoted by horizontal red lines agree with the numerical values (blue
dots). (b) Even $m$ $\omega_{1}=1,\ \omega_{2}=2$: The expected values (family
I:red and green horizontal lines, Family II: only green horizontal lines)
agree with the numerical values, and the common values have larger degeneracy.
(c) Even $m$ and odd $n$ $\omega_{1}=1,\ \omega_{2}=\frac{2}{5}$: The expected
values (family I:red and green horizontal lines, Family II: only green
horizontal lines) agree with the numerical values, and the common values have
larger degeneracy.

Next we use Weyl’s law to validate our computations of correspondence between
the classical families of periodic orbits and the energy levels. Recall that
for the two dimensional case, Weyl’s law is:

$\displaystyle N[E_{j}:E_{j}\leq b]=\frac{1}{\hbar^{2}}\text{Vol}(H\leq
b)+o(1)\text{ as }\hbar\rightarrow 0$ (24)

and notice that the phase space volume for the step-oscillator is:

$\displaystyle\text{Vol}(\mathcal{E})=\int_{0}^{I_{2}(\mathcal{E})}dI_{2}\int_{0}^{I_{1}(\mathcal{E}-E(I_{2})))}-4\theta_{1}^{wall}(I_{1})\theta_{2}^{wall}(I_{2})$
(25)
$\displaystyle+4\pi(\theta_{1}^{wall}(I_{1})+\theta_{2}^{wall}(I_{2}))dI_{1}.$

For the case of a step at the origin and harmonic oscillators, we obtain

$\displaystyle\text{Vol}(\mathcal{E})|_{S_{0},\text{Harmonic
oscillators}}=\frac{3\pi^{2}}{2\omega_{1}\omega_{2}}\mathcal{E}^{2}.$ (26)

Fig. 3 shows this expected correspondence. For the even $m$ case the
contribution of the larger degeneracy associated with the energy levels which
are common to the 2 different families is evident.

.

Figure 3: Weyl’s law. Smooth curves correspond to the predicted phase space
volume (Eq.26) for the three resonant cases ($\omega_{1}=1,\omega_{2}=1,2,3$
yellow, red, blue lines respectively). These prediction fit the corresponding
numerical results. The inset shows the non-uniform jump in $N$ for the even
$m$ case.

###  3.2 Wavefunctions structure and eigenvalues statistics for quantum step
oscillators

We examine first non-resonant oscillators (and not necessarily harmonic) while
keeping the step at the origin. Classically, the motion is ergodic within the
level set for almost all partial energies. Hence, we expect wavefunctions to
concentrate on the projection of such level sets to the configuration space.
We show that at least for a sequence of density $\frac{1}{3}$ of the
wavefunctions this property holds and doesn’t vanish at high energies.
In the correspondent smooth system the potential,$V=V_{1}(q_{1})+V_{2}(q_{2})$
is separable. Thus, its wavefunctions, $\Psi^{sm}_{n}$, can be written as a
product of the wavefunctions of $H_{i}$:
$\\{\Psi^{sm}_{n}\\}_{n=1}^{\infty}=\Psi_{1,k_{1}}(q_{1})\Psi_{2,k_{2}}(q_{2})$
where $\\{\Psi_{i,k_{i}}\\}_{k_{i}=1}^{\infty}$ are the wavefunctions of the
smooth one dimensional Hamiltonian $H_{i}$ and
$E^{sm}_{n(k_{1},k_{2})}=E_{k_{1}}+E_{k_{2}}$.
Since $V_{i}$ are even:

$\displaystyle\Psi_{i,k_{i}}(q_{i})=\begin{cases}\Psi_{i,k_{i}}(-q_{i})\phantom{-}\text{if
$k_{i}$ is even}\\\ -\Psi_{i,k_{i}}(-q_{i})\phantom{-}\text{if $k_{i}$ is
odd}\end{cases}$ (27)

When both $k_{1}$ and $k_{2}$ are odd, the series of wavefunctions
$\\{\Psi^{sm}_{n_{j}(k_{1},k_{2})}\\}_{n_{j}=1}^{\infty}$ vanishes on both
axes, hence, the non-smooth Hamiltonian for the case of step at the origin has
a subsequence of wavefunctions of the form:

$\displaystyle\Psi^{S_{0}}_{n_{j}(k_{1},k_{2})}(q_{1},q_{2})=\begin{cases}\begin{split}\Psi^{sm}_{\tilde{n}_{j}(k_{1},k_{2})}=&\Psi_{1,k_{1}}(q_{1})\Psi_{2,k_{2}}(q_{2}),\\\
&(q_{1},q_{2})\in\mathbb{R}^{2}/S_{0}\end{split}\\\ \\\
\phantom{-}\phantom{-}0\phantom{-}&(q_{1},q_{2})\in S_{0}\end{cases}$ (28)

These solutions are smooth in the domain ($\mathbb{R}^{2}/S_{0}$) and satisfy
Dirichlet boundary conditions on $S_{0}$. Moreover,
$\Psi^{S_{0}}_{n_{j}(k_{1},k_{2})}$ concentrates on the projection of
classical level sets; as the one-dimensional wavefunctions are well
approximated by the WKB approximation [9], they decay exponentially outside of
the classical allowed region of motion:

$\displaystyle\displaystyle\Psi_{i,k_{i}}(q_{i})\approx
C_{0}{\frac{e^{\theta+i\hbar^{-1}\int{\sqrt{2\left(E_{i,k_{i}}-V_{i}(q_{i})\right)}}\,dq_{i}}}{\hbar^{-1/2}{\sqrt[4]{2\left(E_{i,k_{i}}-V_{i}(q_{i}))\right)}}}}.$
(29)

Next we show that the fraction of such odd wavefunctions for the case of a
step at the origin is $1/3$. From equation 22 for the smooth case (i.e. $b=0$)
we deduce that wavefunction that are odd in both directions (odd
$k_{1},k_{2}$) constitute one quarter of all wavefunctions:

$\displaystyle\lim_{E\to\infty}\frac{\\#\\{\Psi^{sm}_{\tilde{n}_{j}(k_{1},k_{2})}:E_{\tilde{n}_{j}}=E^{1}_{k_{1}}+E^{2}_{k_{2}}\leq
E\\}}{\\#\\{\Psi^{sm}_{n}:E_{n}\leq E\\}}=\frac{1}{4}.$ (30)

Since the step is at the origin:

$\displaystyle\text{Vol}(E)^{S_{0}}=\frac{3}{4}\text{Vol}(E)^{sm}$ (31)

and thus, by Weyl’s law

$\displaystyle\lim_{E\to\infty}\frac{\\#\\{\Psi^{S_{0}}_{n_{j}}:E_{n_{j}}\leq
E\\}}{\\#\\{\Psi^{S_{0}}_{n}:E_{n}\leq
E\\}}=\lim_{E\to\infty}\frac{\\#\\{\Psi^{sm}_{\tilde{n}_{j}}:E_{\tilde{n}_{j}}\leq
E\\}}{\frac{3}{4}\\#\\{\Psi^{sm}_{n}:E_{n}\leq E\\}}=\frac{1}{3}.$ (32)

We conclude that for a step at the origin there is no quantum ergodicity in
configuration space, and, in fact, there is a positive measure set of
eigenfunctions that concentrate on the classical level sets.

To examine the behavior for non-symmetric pseudointegrable cases, we study
numerically the shifted corner in the harmonic case: we find the level spacing
of the eigenvalues and study the projections to configuration space of the
eigenfunctions. Both studies propose that the shift does not break the
concentration of a large subset of eigenfunctions on classical level sets.

It is convenient for the study of the non-symmetric system to keep the step at
the origin and shift the original harmonic potential to have a minimum at
$(\epsilon_{1},\frac{\epsilon_{2}}{\omega_{2}^{2}})=\epsilon\cdot(\cos\alpha,sin\alpha)$.
Then the potential is of the form: $V=U_{0}+U_{1}$ where
$U_{0}=\frac{q_{1}^{2}}{2}+\frac{\omega_{2}^{2}q_{2}^{2}}{2}$ and
$U_{1}=-\epsilon_{1}q_{1}-\epsilon_{2}q_{2}$. Here, $\epsilon=0$ corresponds
to the system with a step at the origin, and we study the behavior for a non-
resonant case at finite values of $\epsilon$, beyond the small perturbation
regime. Figure 4(a) compares the cumulative mean level spacing distribution of
this shifted potential of the first 1500 energy levels to the cumulative
Poisson distribution (characterizing integrable systems, $N_{p}(s)=1-e^{-s}$,
reflecting their locality in the classical phase space) and to the cumulative
random matrix ensembles distribution, GOE (characterizing chaotic systems,
$N_{W}(s)=1-e^{-\frac{\pi s^{2}}{4}}$, reflecting their non-local nature in
the classical phase space). We obtain intermediate statistics as in pseudo
integrable billiards, close to semi-Poisson distribution
($N_{sp}(s)=1-e^{-2s}(2s+1)$) [7] (such a behaviour was also observed in a
certain range of parameters in step-like time dependent one d.o.f. Hamiltonian
[14]).

Figure 4(a) shows that the dependence of the level spacing on $\epsilon$
appears to be mild and similar to the case $\epsilon=0$. Recall that in the
case of a step at the origin, we showed that there is a positive density
sequence of eigenfunctions concentrated on classical level sets. Namely, the
level spacing distribution at $\epsilon=0$ reflects this locality in phase
space, together with the non-locality associated with pseudointegrability.
Fig. 4(a) suggests that this behaviour persists when the step is shifted from
the origin. In fact, Fig. 4(b) shows that the distribution with the largest
repulsion is achieved at $\epsilon=0$.

(a)

(b)

Figure 4: PDF and CDF of the level spacing for a non-resonant Hamiltonian for
several positions of the step. The semi-Poisson distribution (solid thick
green line) provides the best fit for all positions of the step (dashed
lines), including a step at the origin (blue dashed line). (a) Cumulative
distribution functions of Poisson, semi-poisson GOE and numerically calculated
CDFs (b) Probability density functions of Poisson, semi-poisson GOE and
numerically calculated PDFs. The level spacing are found by a finite
differences scheme for the time independent Schrodinger Eq. for the
Hamiltonian 20 with $V=U_{0}+U_{1}$ where
$U_{0}=\frac{q_{1}^{2}}{2}+\frac{\omega_{2}^{2}q_{2}^{2}}{2}$ and
$U_{1}=-\epsilon_{1}q_{1}-\epsilon_{2}q_{2}$. The step is located at the
origin and is numerically represented as $V=10^{28}$. Here,
$\omega_{1}=1,\omega_{2}=\sqrt{2}$ and
$(\epsilon_{1},\epsilon_{2})=(0,0),(0.5,0.25),(1,0.5),(1.5,0.75),(\sqrt{3},\frac{\sqrt{3}}{2})$.

To substantiate the claim that, as suggested by the level spacing plots, at
large energies, the general step system still has a positive fraction of
wavefunctions that concentrate on classical level sets, we calculate the
wavefunctions for such systems. Since the wavefunctions depend continuously on
$\epsilon$, for any given maximal energy, for small enough $\epsilon$, such a
fraction of concentrated wavefunctions exists. Hence, we first find the
natural scaling of $\epsilon$ with $E$ and establish that our wavefunction
calculations are far from the trivial limit of ${\epsilon\to 0}$, namely, that
the perturbed wavefunctions do not correlate well with unperturbed
wavefunctions.
Expanding the wavefunctions in $\epsilon$, the first order correction to
$|n(\epsilon)\rangle=|n^{(0)}\rangle+\epsilon|n^{(1)}\rangle+O(\epsilon^{2})$,
is:
${\displaystyle\epsilon|n^{(1)}\rangle=\sum_{k\neq n}{\frac{\langle
k^{(0)}|U_{1}|n^{(0)}\rangle}{E_{n}^{(0)}-E_{k}^{(0)}}}|k^{(0)}\rangle}\newline
$where$\phantom{-}U_{1}=-\epsilon_{1}q_{1}-\epsilon_{2}q_{2}$. So for large
energies, the number and power of terms that contribute significantly to the
sum are expected to stabilize provided we use the scaling:
$\epsilon_{1}\propto\frac{E_{n+1}-E_{n}}{q_{1}}$ and
$\epsilon_{2}\propto\frac{E_{n+1}-E_{n}}{q_{2}}$. Since, for harmonic
oscillators, $q_{i}\propto\sqrt{E}$ and $N(E)\propto\text{Vol}(E)\propto
E^{2}$, so $E_{n+1}-E_{n}\propto\frac{1}{E}$, we conclude that the
stabilization is achieved provided $\epsilon\propto\frac{1}{E^{1.5}}$. As
higher orders of the perturbation series give the same result, we actually
expect that
${\displaystyle|n(\epsilon)}\rangle-{\displaystyle|n^{(0)}\rangle}=O(\epsilon
E_{n}^{1.5})$. To capture the distance between eigenfunctions of the non-
perturbed Hamiltonian to the perturbed one around an energy level $E_{N}$, we
calculate $P$, the mean squared maximal projection on unperturbed
wavefunctions, and $T$, the mean number of above-threshold contributing
unperturbed wavefunctions:

$\begin{split}P(\epsilon,N;\Delta N,J)&=\frac{1}{\Delta N}\sum_{n=N}^{N+\Delta
N}\max_{j^{0}\leq J}{{|\langle\displaystyle
j^{0}\displaystyle|n(\epsilon)}\rangle}|^{2}\\\ T(\epsilon,N;\Delta
N,J,\delta)&=\frac{\sum_{n=N}^{N+\Delta N}\\#({{|\langle\displaystyle
j^{0}\displaystyle|n(\epsilon)}\rangle|}^{2}>\delta)}{\sum_{n=N}^{N+\Delta
N}\sum_{j^{0}=0}^{J}{\langle\displaystyle
j^{0}\displaystyle|n(\epsilon)}\rangle^{2}}.\end{split}$ (33)

Figure 5 shows that $P(\epsilon E_{N}^{3/2},N;\Delta N,J)$ and $T(\epsilon
E_{N}^{3/2},N;\Delta N,J,\delta)$ are, to a good approximation, independent of
$N$, supporting the validity of our scaling. Moreover, while for small
$\epsilon(\frac{E_{N}}{E_{301}})^{3/2}$ we see that, as expected, there is a
strong correlation between the perturbed and unperturbed wavefunctions, for
$\epsilon E_{N}^{3/2}\geq E_{301}^{3/2}$ the maximal projection, $P$, is small
while the level of mixing, $T$, is large, indicating that for such values of
$\epsilon E^{3/2}$ we are indeed far from the small $\epsilon$ limit.
Additional computations show that a further increase in $\epsilon E_{N}^{3/2}$
leads to further decrease in $P$.

(a)

(b)

Figure 5: Scaling of the perturbed wavefunctions with $\epsilon$ and energy.
(a) The mean maximal projection on unperturbed wavefunctions along the energy-
scaled $\epsilon$, $\epsilon(\frac{E_{N}}{E_{301}})^{3/2}$:
$P(\epsilon(\frac{E_{N}}{E_{301}})^{3/2},N;10,400)$ (b) The mean number of
above-threshold contributing unperturbed wavefunctions along the energy-scaled
$\epsilon$, $\epsilon(\frac{E_{N}}{E_{301}})^{3/2}$:
$T(\epsilon(\frac{E_{N}}{E_{301}})^{3/2},N;10,400,0.01)$. These functions are
plotted for $N=151,201,251,301$ and for several
$\epsilon=(\epsilon_{1},\epsilon_{2}=\frac{\epsilon_{1}}{2})$ values.

Finally, we show that even when $\epsilon(\frac{E_{n}}{E_{301}})^{3/2}\gg 1$,
i.e. when the wavefunctions are not well approximated by the unperturbed
wavefunctions, a substantial fraction of the wavefunctions concentrate on
classical level sets. Figure 6 shows the 1481-1500 wavefunctions in
Logarithmic scale normalized by the maximal absolute value of the
wavefunctions for the unperturbed (step at the origin) and perturbed
($\epsilon=(1.5,0.75)$) wavefuncations (so
$\epsilon(\frac{E_{1500}}{E_{301}})^{3/2}=5.25$). For both the perturbed and
unperturbed systems, wavefunctions that are concentrated along the classical
level sets, i.e., are essentially restricted to the configuration space region
$(q_{1},q_{2})\in[q_{1}^{min}(E_{1},\epsilon_{1}),q_{1}^{max}(E_{1},\epsilon_{1})]\times[q_{2}^{min}(E_{2},\epsilon_{2},\omega_{2}),q_{2}^{max}(E_{2},\epsilon_{2},\omega_{2})]\setminus
S_{q^{wall}}$ where $q_{i}^{max,min}$ correspond to the classical level set
boundaries, are clearly seen (e.g. see wavefunction 1 in the unperturbed
system and wavefunction 19 in the pertubed system). We call such wavefunctions
concentrated wavefunctions.

(a)

(b)

Figure 6: High energy wavefunctions for a step at the origin and for a shifted
step. (a) The unperturbed Hamiltonian. (b) The perturb Hamiltonian with
$(\epsilon_{1},\epsilon_{2})=(1.5,0.75)$. The wavefunctions for $n=1481-1500$
are plotted. To better visualize the main mass concentration we plot
$Log(|\Psi_{n}(q_{1},q_{2};\epsilon)|+\max_{q_{1},q_{2}}|\Psi_{n}((q_{1},q_{2};\epsilon)|)$.

To quantify this observation, we need to distinguish between concentrated
wavefunctions from wavefunctions which are not concentrated. To this aim we
define vertical and horizontal means of the wavefunctions:

$\begin{split}M^{H}_{n}(q_{2})&=\int_{-\infty}^{\infty}|\Psi_{n}(q_{1},q_{2})|^{2}dq_{1}\\\
M^{V}_{n}(q_{1})&=\int_{-\infty}^{\infty}|\Psi_{n}(q_{1},q_{2})|^{2}dq_{2}.\end{split}$
(34)

and suggest that

$\tilde{E}=\frac{V_{1}(arg\max_{q_{1}}M^{V}_{n}(q_{1}))+V_{2}(arg\max_{q_{2}}M^{H}_{n}(q_{2})}{E}$
(35)

provides a good indicator for the wavefunctions concentration: it is close to
one for concentrated wavefunctions and has a much lower value for the rest of
the wavefunctions.

Figures 7(a,b) present $\tilde{E}$ values in the case of corner at the origin
for low (a) and high (b) ranges of energies. Red points represent $\tilde{E}$
values for the product wavefunctions of Eq. (28) and constitute around $1/3$
of the 20 $\tilde{E}$ values. We see that some of the blue points align with
the red ones, while others, around 1/5 for the lower energies and 1/2 for the
higher energies have a much lower value. The insets present
$(M^{H}_{n},M^{V}_{n})$ in the positive quadrant for the three different types
of wavefunctions: for a product wavefunction (red point, wavefunction 1 in
7(a) ), for a concentrated wavefunction with a similar $\tilde{E}$ value (blue
point, wavefunction 13 in 7(a) ) and for a non-concentrated wavefunction with
a low $\tilde{E}$ value (blue point, wavefunction 9 in 7(a) ). In the first
two cases we recognize an oscillatory structure within the classically allowed
region, and we observe that the maximal power appears close to the edge. In
contrast, the insets corresponding to the low $\tilde{E}$ value show a non
oscillatory structure with peaks at arbitrary positions within the Hill
region.
Figures 7(c,d) present a similar computation for the case of the shifted
potential, $\epsilon=(1.5,0.75)$, for which there are no product
wavefunctions, yet concentrated and not concentrated wavefunction do appear,
and the indicator $\tilde{E}$ seems to distinguish between these two types of
wavefunctions.

The reasoning for this suggestion is as follows; For step at the origin, for
the product wavefunctions (eq. 28), $M^{H}_{n}(q_{2})=|\Psi_{n,2}(q_{2})|^{2}$
for $q_{2}>0$ and $M^{H}_{n}(q_{2})=|\Psi_{n,2}(q_{2})|^{2}/2$ for $q_{2}<0$,
so by the WKB approximation (eq.29), and similarly for $M^{V}_{n}(q_{1})$, we
indeed expect $\tilde{E}=1-f(E)$ for some function $f(E)$ which tends to zero
as $E$ goes to infinity (e.g., Figures 7(a,b) suggest that $f(E_{500})\approx
0.15,E_{500}=39.9$ and $f(E_{1500})\approx 0.1,E_{1500}=70.5$). For non-
product yet concentrated wavefunctions on some classical configuration space
region defined by the partial energies $(E_{1},E_{2})$, the $argmax$ of
$M^{V,H}$ cannot be larger than the corresponding $q_{i}^{max}$. Moreover, as
classically, one of the momenta components vanishes at the edges of the
classical region, the projection of the Liuoville measure to the configuration
space there is expected to be larger, hence, by the correspondence principle,
we expect maximal densities near the edges. Hence, $\tilde{E}$ provides the
approximate ratio between the sum of the potential energies at the classical
region corners (belonging to the boundary of the classical Hill region) to the
total energy, so we expect it to have a similar $\tilde{E}$ values to the
corresponding product wavefunctions. In contrast, for a wavefunction which
does not concentrate on a single classical level set we do not expect the
maxima in the horizontal and vertical directions to lie necessarily on the
boundary of the Hill region (see insets corresponding to the lower $\tilde{E}$
values), thus the sum of the potential energies at such an interior point
leads to a lower value of $\tilde{E}$.

In conclusion, Figures 6 and 7 suggest that the fraction of concentrated
wavefunctions does not vanish at high energies even when the step is shifted.

Figure 7: An indicator for the concentration of wavefunctions on classical
level sets. The indicator $\tilde{E}$ of Eq. 35 is plotted for the case of a
step at the origin, (a) 481-500 and (b) 1481-1500 wavefunctions (the
wavefunctions of Fig. 6a). The indicator $\tilde{E}$ is plotted for the case
of a shifted step ( $\epsilon=(1.5,0.75)$) (c) 481-500 and (d) 1481-1500
wavefunctions(the wavefunctions of Fig. 6b).
The insets present $M^{H},M^{V}$ for specific points

##  4 Summary and discussion

We studied the correspondence of a quantum step-oscillator - a two dimensional
quantum oscillator in the presence of a step (a step-like region $S$ in the
configuration space at which the potential energy is infinite) to its
classical analog, a pseudointegrable Hamiltonian impact system. For the case
of harmonic resonant oscillators with a corner at the origin, for which
families of periodic orbits can be explicitly constructed, we demonstrated
that the EBK quantization condition provides a good predictor to the energy
levels (Figure 8), and that Weyl’s law provides a good approximation to the
growth in the number of wavefunctions (Figure 3). Moreover, we observed that
in even-resonance cases two different families of periodic orbits belonging to
the same component of the level set co-exist, with distinct corresponding
wavefunctions, each contributing a positive portion to the phase space volume
(Figure 3). This demonstrates that the non-ergodicity of level sets has a
quantum analog. We showed that the intermediate level spacing of the quantum
step-oscillator for non-resonant and not necessarily harmonic potential hardly
depends on the position of the step (taken in the negative quadrant) and is
approximately semi-Poisson, indicating repulsion of energy levels, similar to
the level spacing obtained for pseudointegrable billiards (Figure 4). When the
step is at the origin, we showed that there is a positive fraction of
wavefunctions that remain concentrated along the classical level sets at
arbitrarily high energies, as occurs for integrable systems, namely they do
not tend to equidistribute in the configuration space as is the case for
pseudointegrable billiards (Eq. (28)-(32) and Figures 6a and 7a,b). Finally,
when the corner is shifted from the origin, we conjecture, based on numerical
evidence for non-resonant harmonic oscillators, that there is a positive
density series of wavefunctions which are not equidistributed and concentrated
along the classical level sets (Figures 6b and 7c,d).

##  5 appendix

######  Definition 5.1.

A family of periodic orbits of the step system is the family of periodic
orbits having identical number of turning points and impact points $b_{i}$ and
$\mu_{i}$

######  Proposition 5.2.

For a step at the origin, on level sets with
$\omega_{1}=1,\omega_{2}=\frac{1}{n}$ the step system has:

1) For an odd $n$: exactly one family of p.o. with
$b_{1}=n,\mu_{1}=3n,b_{2}=1,\mu_{2}=3$ and action
$I=\frac{3nI_{1}+3I_{2}}{2}$. Its quantization condition is:
$E_{(k)}=\frac{k}{1.5n}+\frac{5(1+n)}{6n}$
2) For an even $n$: exactly two families of p.o. one with $\mu^{I}_{1}=2n,$
$b^{I}_{1}=n$, $\mu^{I}_{2}=2$,$b^{I}_{2}=0$ and action
$I^{I}=\frac{2nI_{1}+2I_{2}}{2}$ and another one with
$\mu^{II}_{1}=n,\mu^{II}_{2}=1,b^{II}_{1}=0,b^{II}_{2}=1$ and action
$I^{II}=\frac{nI_{1}+I_{2}}{2}$. Theirs quantization condition:
$E^{I}_{(k)}=\frac{k}{n}+\frac{4n+2}{4n}$,
$E^{II}_{(k)}=\frac{2k}{n}+\frac{n+3}{2n}$

###### Proof.

Consider the Poincare map restricted to a given level set,
$I(H_{1}=e_{1},H_{2}=E-e_{2})$,
$P_{1}:\Sigma_{1}^{wall}\rightarrow\Sigma_{1}^{wall}$, so
$P_{1}:\theta_{2}\rightarrow\bar{\theta}_{2},\theta_{2}\in[0,2\pi)$. Divide
this section to $2n$ equal length intervals starting from $\frac{\pi}{2}$ (so
$s_{i}=(\frac{\pi}{2}+(i-1)\frac{\pi}{n},\frac{\pi}{2}+i\frac{\pi}{n}\text{
mod }2\pi),\ i=1,...,2n$). Since $\omega_{1}=1,\omega_{2}=\frac{1}{n}$ and the
step is at the origin (so the width of each strip in the cross is $\pi$), the
transition between these segments and the corfresponding increase in the
turning points and impacts is (notice that each crossing of the section
$\theta_{i}=0$ or $\theta_{i}=\pi$ corresponds to an additional turning point
in the $q_{i}$ direction):
$\begin{cases}s_{i}\rightarrow
s_{i+1},(b_{1},\mu_{1},b_{2},\mu_{2})\rightarrow(b_{1}+1,\mu_{1}+1,b_{2},\mu_{2})&\text{if:
$i<n-1$ , $i\neq\left\lfloor{\frac{n}{2}}\right\rfloor$ or $i=2n$}\\\
s_{i}\rightarrow
s_{i+1},(b_{1},\mu_{1},b_{2},\mu_{2})\rightarrow(b_{1}+1,\mu_{1}+1,b_{2},\mu_{2}+1)&\text{if:
$i=\left\lfloor{\frac{n}{2}}\right\rfloor$}\\\ s_{i}\rightarrow
s_{i+2},(b_{1},\mu_{1},b_{2},\mu_{2})\rightarrow(b_{1},\mu_{1}+2,b_{2},\mu_{2})&\text{if:$n\leq
i<2n-1,i\neq\left\lfloor{\frac{3n}{2}}\right\rfloor,\left\lfloor{\frac{3n}{2}}\right\rfloor-1$}\\\
s_{i}\rightarrow
s_{i+2},(b_{1},\mu_{1},b_{2},\mu_{2})\rightarrow(b_{1},\mu_{1}+2,b_{2},\mu_{2}+1)&\text{if:
$i=\left\lfloor{\frac{3n}{2}}\right\rfloor,i=\left\lfloor{\frac{3n}{2}}\right\rfloor-1$}\\\
s_{i}\rightarrow
s_{n+1},(b_{1},\mu_{1},b_{2},\mu_{2})\rightarrow(b_{1},\mu_{1}+2,b_{2}+1,\mu_{2})&\text{if:
$i=2n-1$}\end{cases}$

Thus, orbits starting at $s_{1}$ always reach $s_{n}$ after $n$ iterations,
undergoing up to this point $n-1$ impacts with the right wall, $n$ turning
points in the $q_{1}$ direction ($n$ crossing of the section $\theta_{1}=0$)
and $1$ turning point in the $q_{2}$ direction.

For an even $n$, $s_{n}$ is mapped to $s_{2n}$ after $\frac{n}{2}$ iterations
without passing through $s_{2n-1}$, and then maps back to $s_{1}$. Namely,
such orbits complete a period by visiting all segments $s_{1},..,s_{n}$ and
only even segments between $s_{n+1}$ and $s_{2n}$. Thus, $\mu^{I}_{1}=2n,$
$b^{I}_{1}=n$, $\mu^{I}_{2}=2$, $b^{I}_{2}=0$.

The other family of orbits starts at $s_{n+1}$ and visits only odd segments
between $s_{n+1}$ and $s_{2n}$. Thus, it undergoes a single impact with the
upper wall, has $n$ turning points in the $q_{1}$ direction and a single
turning point in the $q_{2}$ direction:
$\mu^{II}_{1}=n,\mu^{II}_{2}=1,b^{II}_{1}=0,b^{II}_{2}=1$.

For an odd $n$, $s_{n}$ is mapped to $s_{2n-1}$ after
$\lfloor{\frac{n}{2}}\rfloor$ iterations, then impacts the upper wall and maps
to $s_{n+1}$. Then it visits only even segments between $s_{n+1}$ and $s_{2n}$
and maps back to $s_{1}$. Thus, $b_{1}=n,\mu_{1}=3n,b_{2}=1,\mu_{2}=3$ ∎

Figure 8: Trajectories of family I(green) and family II(red) in action angle
coordinates for $\omega=1$, $\omega=\frac{1}{2}$ with the c-series $[0;2]$.
Proposition 5.2 states: $(\mu^{I},b^{I})=((4,2),(2,0))$ and
$(\mu^{II},b^{II})=((2,1),(0,1))$. The four different background color stands
for the 4 $s_{i}$ segments.

Now consider $\omega_{1}=1,\omega_{2}=\frac{m}{n}$:

######  Definition 5.3.

c-series: Given a rational number $\frac{m}{n}=\frac{m_{0}}{n_{0}}$ the
associated even continued fraction series, $[c_{0};c_{1},...,c_{k}]$ and its
partial expansions, $\frac{m_{i}}{n_{i}},i=0,..,k$ are defined inductively on
$i$; Given $\frac{m_{i}}{n_{i}}$ and $[c_{0};c_{1},...,c_{i-1}]$ (so at $i=0$
the previous c-series is empty) define:

  * •

If $\frac{m_{i}}{n_{i}}$ is an integer then $c_{i}=\frac{m_{i}}{n_{i}}$ and
the series ends, so $k=i$.

  * •

Otherwise,

$c_{i}=\left\lfloor{\frac{m_{i}}{n_{i}}}\right\rfloor+N$

where

$N=\begin{cases}1,&\text{if:}\left\lfloor{\frac{m_{i}}{n_{i}}}\right\rfloor\text{is
odd }\\\ 0,&\text{if:}\left\lfloor{\frac{m_{i}}{n_{i}}}\right\rfloor\text{is
even}\end{cases}$

and

$\frac{m_{i+1}}{n_{i+1}}=(\frac{m_{i}}{n_{i}}-c_{i})^{-1}$

where $n_{i+1}$ always gets the positive sign.

######  Lemma 5.4.

A rational number $\frac{m_{i}}{n_{i}}$ associated with a c-series
$[c_{i};...,c_{k}]$ is equal to $c_{i}+\frac{n_{i+1}}{m_{i+1}}$ where
$\frac{m_{i+1}}{n_{i+1}}$ is associated with $[c_{i+1};...,c_{k}]$

###### Proof.

Derives directly from the definition. ∎

######  Lemma 5.5.

The c-series is finite and unique, $k=0$ iff $\frac{m}{n}$ is an integer, and,
for all $k>0$ and $0\leq i\leq k-1$, $c_{i}$ is even, whereas $c_{k}$ can be
either even or odd. For all $i$, $c_{i}$ can be negative, $c_{k}\neq 1$, and
for all $i>0,\ c_{i}\neq 0$.

###### Proof.

We use proof by induction: If $k=0$, $\frac{m}{n}$ is an integer number and
the c-series ,$[c_{k=0}=\frac{m}{n}]$, is finite and $c_{k}$ can be even or
odd.
Otherwise, assume - at the $i^{\prime}th>0$ step, given a non integer rational
number $\frac{m_{i}}{n_{i}}$, and a sequence $[c_{i};c_{i+1},...,c_{k}]$, we
compute, $\frac{m_{i+1}}{n_{i+1}}=\begin{cases}-(\frac{n_{i}-(m_{i}\mod
n_{i})}{n_{i}})^{-1},&\text{if:}\left\lfloor{\frac{m_{i}}{n_{i}}}\right\rfloor\text{is
odd }\\\ (\frac{m_{i}\mod n_{i}}{n_{i}})^{-1},&\text{
if:}\left\lfloor{\frac{m_{i}}{n_{i}}}\right\rfloor\text{is even}\end{cases}$
where $n_{i+1}$ always gets the positive sign.
Thus, $|m_{i+1}|=n_{i}$
And, $n_{i+1}<n_{i}$
As a result of that after a finite number of steps $t\leq n_{0}$, $n_{i+t}=1$,
$\frac{m_{i+t}}{n_{i+t}}$ is an integer and the series is finite.
In addition: $|\frac{m_{i+1}}{n_{i+1}}|>1$, which implies $\ c_{i}\neq 0$
It follows that for any given $\frac{m_{0}}{n_{0}}$ the process defines all
$m_{i}$, $n_{i}$ and $c_{i}$ uniquely. ∎

######  Lemma 5.6.

given $\frac{m_{i}}{n_{i}}$ associated with $[c_{i};...,c_{k}]$ and
$\frac{m_{i+1}}{n_{i+1}}$ associated with $[c_{i+1};...,c_{k}]$, the eveness
properties of $(m_{i+1},n_{i+1})$ are identical to those of $(n_{i},m_{i})$,
namely, $m_{i+1}\mod 2=n_{i}\mod 2$ and $n_{i+1}\mod 2=m_{i}\mod 2$.

###### Proof.

by definition

$\frac{m_{i+1}}{n_{i+1}}=(\frac{m_{i}}{n_{i}}-c_{i})^{-1}$

where $c_{i}$ is even:
Thus, $|n_{i+1}|\mod 2=|m_{i}-c_{i}n_{i}|\mod 2=m_{i}\mod 2$
$|m_{i+1}|\mod 2=|n_{i}|\mod 2=n_{i}\mod 2$ ∎

######  Lemma 5.7.

given $\frac{m_{0}}{n_{0}}$ associated with $[c_{0};...,c_{k}]$
1) if $m_{0}$ is even and $n_{0}$ is odd then $c_{k}$ is even and k is even
2) if $m_{0}$ is odd and $n_{0}$ is even then $c_{k}$ is even and k is odd
3) if $m_{0}$ is odd and $n_{0}$ is odd then $c_{k}$ is odd

###### Proof.

$c_{k}=\frac{m_{k}}{n_{k}}$
1) For an even $m_{0}$ and an odd $n_{0}$ we assume by contradiction an odd k:
by lemma 5.6 $m_{k}$ is odd and $n_{k}$ is even. However,
$c_{k}=\frac{m_{k}}{n_{k}}$ is an integer, which creates a contradiction.
Thus, the only possible option is an even k, an even $m_{k}$ and an odd
$n_{k}(=1)$ which implies an even $c_{k}$.

2) For an odd $m_{0}$ and an even $n_{0}$ we assume by contradiction an even
k: by lemma 5.6 $m_{k}$ is odd and $n_{k}$ is even. However,
$c_{k}=\frac{m_{k}}{n_{k}}$ is an integer, which creates a contradiction.
Thus, the only possible option is an odd k, an even $m_{k}$ and an odd
$n_{k}(=1)$ which implies an even $c_{k}$.
3)By lemma 5.6 for an odd $m_{0}$ and an odd $n_{0}$, $m_{k}$ is odd and
$n_{k}$ is odd. Thus $c_{k}$ is odd as well. ∎

Examples: $\frac{1}{3}=[0,3]_{c}$ $\frac{3}{5}=[0,2,-3]_{c}$
$\frac{2}{5}=[0,2,2]_{c}$ $\frac{29}{12}=[2,2,2,2]$
$\frac{9}{8}=[2,-2,2,-2,2,-2,2,-2]_{c}$

######  Lemma 5.8.

: The c-series is identical to the continued fraction series iff the continued
fraction is of the form $[a_{0};a_{1}...,a_{k}]$ where $a_{i}$ is even for
$0<i<k-1$ and $a_{k}$ can be either even or odd.

###### Proof.

derives directly from the definition. ∎

######  Definition 5.9.

The right impact interval, $J_{r}^{\omega_{2}}$,(For a level set with
$\omega_{1}=1$, $\omega_{2}=\omega_{2}$) is the set of all $\theta_{2}$ at
$\Sigma_{1}^{wall}$, which are mapped by the flow to the step region, namely,
satisfying $|\theta_{2}+2\theta_{1}^{wall}\frac{\omega_{2}}{\omega_{1}}\mod
2\pi|\geq\theta_{2}^{wall}$

######  Lemma 5.10.

The right impact interval of a step system with corner in the origin, with
frequencies $\omega_{2}$ that differ by an even integer are identical: if
$\omega_{1}=1$ and $\omega_{2}=\frac{m}{n}$ and $\omega^{\prime}_{1}=1$ and
$\omega^{\prime}_{2}=\frac{m}{n}+2L$ for an integer $L$ then
$J_{r}^{\frac{m}{n}+2L}=J_{r}^{\frac{m}{n}}$.

###### Proof.

$J_{r}^{\frac{m}{n}}=\\{\theta_{2}:|\theta_{2}+\pi\frac{m}{n}\mod
2\pi|>\theta_{2}^{wall}\\}$
Now since, $|\theta_{2}+\pi\frac{m}{n}+2\pi L\mod
2\pi|=|\theta_{2}+\pi\frac{m}{n}\mod 2\pi|$,
$J_{r}^{\frac{m}{n}}=J_{r}^{\frac{m}{n}+2L}$ ∎

######  Lemma 5.11.

A step system with corner in the origin with $\omega_{1}=1$ and
$\omega_{2}=\frac{m}{n}$ and a step system with corner in the origin with
$\omega^{\prime}_{1}=1$ and $\omega^{\prime}_{2}=\frac{m}{n}+2L$ for an
integer L has the same number of families of p.o

###### Proof.

Given $\omega_{1}=1$ and the rational frequencies $\omega_{2}=\frac{m}{n}$ and
$\omega^{\prime}_{2}=\frac{m}{n}+2L$ the Poincare map
$P_{1}^{\omega_{2}}:\Sigma_{1}^{wall}\rightarrow\Sigma_{1}^{wall}$ satisfies
the relation $P_{1}^{\frac{m}{n}+2L}$=$P_{1}^{\frac{m}{n}}$.This derives from
lemma 5.10, $J_{r}^{\frac{n}{m}}=J_{r}^{\frac{n}{m}+2L}$, realizing that for
$\theta_{1}\in[-\frac{\pi}{2},\frac{\pi}{2}]$(the vertical center of the
cross) the $\theta_{2}$ coordinate is increased by $2\pi L$ whereas for
$\theta_{1}\in[\frac{\pi}{2},\frac{3\pi}{2}]$(the horizontal part of the
cross) the $\theta_{2}$ coordinate is increased by $4\pi L$, so altogether we
see that:
$P_{1}^{\frac{m}{n}+2L}=\begin{cases}P_{1}^{\frac{m}{n}}+2\pi L,&\text{if:
$\theta_{2}\in J_{r}^{\frac{n}{m}}$}\\\ P_{1}^{\frac{m}{n}}+6\pi
L,&\text{else}\end{cases}$
$\mod 2\pi$, the two maps are identical and have the same dynamics ∎

######  Lemma 5.12.

A step system with corner in the origin with $\omega_{1}=1$ and
$\omega_{2}=\frac{m}{n}$ and a step system with corner in the origin with
$\omega^{\prime}_{1}=1$ and $\omega^{\prime}_{2}=\frac{n}{m}$ has the same
number of families of p.o

###### Proof.

Given $\omega_{1}=1$ and the rational frequencies $\omega_{2}=\frac{m}{n}$ and
$\omega^{\prime}_{2}=\frac{n}{m}$ the Poincare map
$P_{i}:\Sigma_{i}^{wall}\rightarrow\Sigma_{i}^{wall}$ satisfies the relation
$P_{1}^{\frac{m}{n}}$=$P_{2}^{\frac{n}{m}}$. This derives strictly from the
definition of $P_{i}$. The number of families of p.o is the number of
different cycles at $P_{i}$ which has to be identical at both cases because of
the above relation. ∎

######  Lemma 5.13.

A step system with corner in the origin with $\omega_{1}=1$ and
$\omega_{2}=\frac{m_{i}}{n_{i}}$ associated with a c-series
$[c_{i};...,c_{k}]$ and a step system with corner in the origin with
$\omega^{\prime}_{1}=1$ and $\omega^{\prime}_{2}=\frac{m_{i+1}}{n_{i+1}}$
associated with a c-series $[c_{i+1};...,c_{k}]$ have the same number of
families of p.o.

###### Proof.

By lemma 5.4: $\frac{m_{i}}{n_{i}}=c_{i}+\frac{m_{i+1}}{n_{i+1}}$. Thus, the
lemma derives directly from lemmas 5.11 and 5.12.
∎

######  Proposition 5.14.

A step system with corner in the origin with $\omega_{1}=1$ and
$\omega_{2}=\frac{m_{0}}{n_{0}}$ associated with a c-series
$[c_{0};...,c_{k}]$ has 2 families of periodic orbits if $c_{k}$ is even and
one if $c_{k}$ is odd.

###### Proof.

By lemma 5.13 a step system with corner in the origin with $\omega_{1}=1$ and
$\omega_{2}=\frac{m_{0}}{n_{0}}$ associated with a c-series
$[c_{0};...,c_{k}]$ and a step system with corner at the origin with
$\omega_{1}=1$ and $\omega_{2}=\frac{1}{c_{k}}$ have the same number of
families of periodic orbits. By proposition 5.14 if $c_{k}$ is even there are
2 families of periodic orbits and if $c_{k}$ is odd there is one. ∎

######  Lemma 5.15.

For a step system with corner in the origin with $\omega_{1}=1$ and
$\omega_{2}=\frac{m_{i}}{n_{i}}$ associated with a c-series
$[c_{i};...,c_{k}]$ with an odd $c_{k}$($m_{i}$ and $n_{i}$ are odd)
$\mu_{2}=3|m_{i}|$ and $\mu_{1}=3n_{i}$

###### Proof.

We prove the claim by induction:
For $i=k$:
We recall that by proposition 5.2, $\mu_{2}=3|c_{k}|,\mu_{1}=3$. Now assume
the lemma holds for $i=k-l$:
then, $\omega_{2}=\frac{m_{i}}{n_{i}}$, thus $\mu_{2}=3|m_{i}|$ and
$\mu_{1}=3n_{i}$.
Now, for $i-1=k-(l+1)$:
$\omega_{2}^{\prime}=\frac{m_{i-1}}{n_{i-1}}$ associated with
$[c_{i-1};...,c_{k}]$.
From lemma 5.4, $\omega_{2}^{\prime}=c_{i-1}+\frac{n_{i}}{m_{i}}$ and
$c_{i-1}$ is even.
Thus, $P^{\omega_{2}^{\prime}}_{1}=P^{\omega_{2}}_{2}$(and vise-versa) and
$\mu_{2}(\omega_{2}^{\prime})=|\mu_{1}(\omega_{2})+3c_{i-1}|=3|n_{i}+c_{i-1}|=3|m_{i-1}|$
and also $\mu_{1}(\omega_{2}^{\prime})=\mu_{2}(\omega_{2})=3|m_{i}|=3n_{i-1}$.
∎

######  Lemma 5.16.

For a step system with corner in the origin with $\omega_{1}=1$ and
$\omega_{2}=\frac{m_{i}}{n_{i}}$ associated with a c-series
$[c_{i};...,c_{k}]$ with an even $c_{k}$($m_{i}\mod 2\neq n_{i}\mod 2$):
$\mu^{I}_{2}=2|m_{i}|$ and $\mu^{I}_{1}=2n_{i}$ and $\mu^{II}_{2}=|m_{i}|$ and
$\mu^{II}_{1}=n_{i}$

###### Proof.

We prove the claim by induction:
For $k=0$:
We recall that by propositon 5.2, $\mu^{I}_{2}=2|c_{k}|$ and
$\mu^{II}_{2}=|c_{k}|$.
Now assume the lemma holds for $i=k-l$:
then, $\omega_{2}=\frac{m_{i}}{n_{i}}$, Thus $\mu^{I}_{2}=2|m_{i}|$ and
$\mu^{I}_{1}=2n_{i}$ and $\mu^{II}_{2}=|m_{i}|$ and $\mu^{II}_{1}=n_{i}$.

Now, for $i-1=k-(l+1)$:
$\omega_{2}^{\prime}=\frac{m_{i-1}}{n_{i-1}}$ associated with
$[c_{i-1};...,c_{k}]$.
By lemma 5.4 $\omega_{2}^{\prime}=c_{i-1}+\frac{n_{i}}{m_{i}}$ and $c_{i-1}$
is even.
Thus by lemmas 5.11 and 5.12,
$P^{\omega_{2}^{\prime}}_{1}=P^{\omega_{2}}_{2}$(and vise-versa) and
$\mu^{I}_{2}(\omega_{2}^{\prime})=|\mu^{I}_{1}(\omega_{2})+2c_{i-1}|=2|n_{i}+c_{i-1}|=2|m_{i-1}|$
and ,similarly, $\mu^{II}_{2}=|m_{i-1}|$. For the same reason,
$\mu^{I}_{1}=2n_{i-1}$ And $\mu^{II}_{1}=n_{i-1}$ ∎

######  Proposition 5.17.

A step system with corner in the origin with $\omega_{1}=1$ and
$\omega_{2}=\frac{m}{n}$ with an odd m and odd n:
$\mu_{1}=3n,\mu_{2}=3m,b_{1}=n,b_{2}=m$

###### Proof.

By lemma 5.15 $\mu_{1}=3n$, Thus at a period, $P_{1}$ goes through 2n
iterations cycle. n of the iterations are for $\theta_{2}\in J_{r}$, thus,
$b_{1}=n$.
n of the iterations are for $\theta_{2}\notin J_{r}$. We can write
$\omega_{2}=\frac{m}{n}=\left\lfloor{\frac{m}{n}}\right\rfloor+m\mod n$. Thus
we can divide the n iterations to $m\mod n$ iterations with
$\left\lfloor{\frac{m}{n}}\right\rfloor+1$ impacts with the upper wall and
$n-m\mod n$ iterations with $\left\lfloor{\frac{m}{n}}\right\rfloor$ impacts
with the upper wall.

Thus, $b_{2}=n\left\lfloor{\frac{m}{n}}\right\rfloor+m\mod n=m$. ∎

######  Proposition 5.18.

Quantized energy levels for Harmonic oscillator with corner in the origin with
$\omega_{1}=1$ and $\omega_{2}=\frac{m}{n}$ for odd m and n are:
$E_{(k)}=\frac{2k}{3n}+\frac{1+\frac{m}{n}}{2}+\frac{n+m}{3n}=\frac{2k}{3n}+\frac{5(m+n)}{6n}$

###### Proof.

Quantization conditions derived from the equation:
$\frac{\mu_{1}I_{1}+\mu_{2}I_{2}}{2}=k+\frac{\mu_{1}+\mu_{2}}{4}+\frac{b_{1}+b_{2}}{2}$
For harmonic oscillator: $I_{i}=\frac{E_{i}}{\omega_{i}}$
1) for an odd m and odd n:
$\frac{3nE_{1}+3m\frac{E_{2}}{\frac{m}{n}}}{2}=k+\frac{3n+3m}{4}+\frac{n+m}{2}$
$E_{(k)}=\frac{k}{1.5n}+\frac{n+\frac{m}{n}}{2n}+\frac{n+m}{3n}=\frac{2k}{3n}+\frac{5(n+m)}{6n}$
∎

Let $b_{s,i}$ and $\mu_{s,i}$, $s\in\\{1,2\\}$, denote the number of
impacts/turning points for a step system with corner in the origin with
$\omega_{1}=1$ and $\omega_{2}=\frac{m_{i}}{n_{i}}$ associated with a c-series
$[c_{i};...,c_{k}]$.

######  Proposition 5.19.

A step system with a corner in the origin with $\omega_{1}=1$ and
$\omega_{2}=\frac{m}{n}$ associated with a c-series $[c_{0};...,c_{k}]$,with
an even m and odd n has 2 families of periodic orbits such that:
$\mu^{I}_{1}=2n,\mu^{I}_{2}=2m,\mu^{II}_{1}=n,\mu^{II}_{2}=m$.
The number of impacts can be calculated backwards inductively as:
$b^{I}_{1,i}=b^{I}_{2,i+1}$,
$b^{I}_{2,i}=b^{I}_{1,i+1}+c_{i}(\frac{\mu^{I}_{1,i}-b^{I}_{1,i}}{2})$, and
$b^{II}_{1,i}=b^{II}_{2,i+1}$,
$b^{II}_{2,i}=b^{II}_{1,i+1}+c_{i}(\frac{\mu^{II}_{1,i}-b^{II}_{1,i}}{2})$

###### Proof.

Recall that by lemma 5.4
$\omega_{2}=\frac{m_{i}}{n_{i}}=\frac{n_{i+1}}{m_{i+1}}+c_{i}$. For
$\omega_{2}^{\prime}=\frac{n_{i+1}}{m_{i+1}}=\frac{1}{\omega_{2,i+1}}$ we know
that: $b^{\prime I}_{1}=b^{I}_{2,i}$, $b^{\prime I}_{2}=b^{I}_{1,i}$, and
$b^{\prime II}_{1}=b^{II}_{2,i}$, $b^{\prime II}_{2}=b^{II}_{1,i}$. Now as
$c_{i}$ is even the step systems with
$\omega_{2}^{\prime}=\frac{n_{i+1}}{m_{i+1}}$ and
$\omega_{2}=\frac{n_{i+1}}{m_{i+1}}+c_{i}$ have the same Poincare map $P_{1}$.
By definition the number of rounds the periodic trajectory makes for both
$\omega_{2}$ and $\omega^{\prime}_{2}$, in the horizontal part of the
cross($\theta_{1}\in[\frac{\pi}{2},\frac{3\pi}{2}]$) is
$\frac{\mu_{1,i}-b_{1,i}}{2}$ and in each such an iteration the $\omega_{2}$
system has $c_{i}$ additional impacts with the upper wall. The number of
impacts with the right wall($b_{1}$) doesn’t change as $P_{1}$ does not
change.

∎

## References

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##  6 Acknowledgments

I would like to express my deepest gratitude to my advisor, prof. Vered Rom-
Kedar for all the help provided, the time invested in my training and the
knowledge I gained during the process. I have had a great time


# Projectional entropy for actions of amenable groups

Michał Prusik  Faculty of Pure and Applied Mathematics
Wroclaw University of Science and Technology
Wybrzeże Wyspiańskiego 27
50-370 Wrocław Poland<EMAIL_ADDRESS>
###### Abstract.

In the current paper we attempt to transfer the notion of the projectional
entropy, originally defined for multidimensional subshifts, to the case of
actions of amenable groups. The main theorem states that if a system is
strongly irreducible the equality of the entropy and the projectional entropy
implies that the system has a product-like structure.

###### Key words and phrases:

entropy, group action, amenable group, subshift

###### 2020 Mathematics Subject Classification:

Primary 37B05, 37B40; Secondary 37B10

##  1\. Introduction

In “Projectional Entropy in Higher Dimensional Shifts of Finite Type” ([1]) A.
Johnson, S. Kass and K. Madden introduced an object called the projectional
entropy. The authors showed several basic properties of this entropy and
proved a theorem relating this notion to the structure of the subshift in the
case of two-dimensional subshifts. In 2010 M. Schraudner extended their
results in [2] to higher dimensional subshifts, by adding an assumption of
mixing type. In the current paper we aim to generalize the notion of the
projectional entropy for actions of amenable groups.

The entropy is one of the main objects of interest in the theory of dynamical
systems. For the case of symbolic dynamics it gives us an information about
the rate of exponential growth of the number of configurations appearing in
the system. That growth is measured when the domain of a configuration expands
simultaneously in every “direction” of the space. The main intuition behind
the projectional entropy is to look only in some directions. Namely, we
restrict the elements of the subshift $X\subset\mathcal{A}^{\mathbb{Z}^{d}}$
(where $\mathcal{A}$ is a fixed alphabet) to a chosen sublattice $L$, where a
sublattice is a subgroup of $\mathbb{Z}^{d}$ isomorphic to $\mathbb{Z}^{r}$
for some $r<d$, which has a complementary sublattice $L^{\prime}$ (namely the
quotient group $\mathbb{Z}^{d}/L$) isomorphic to $\mathbb{Z}^{(d-r)}$. So
obtained restriction $X_{L}$ is treated as an $r$-dimensional subshift and its
entropy is called the $L$-projectional entropy of $X$. By comparing this
quantity with the entropy of the original space $X$ we get some information
about a “product-like” structure of $X$.

It turns out that one does not need to limit only to sublattices. Consider
$L=\mathrm{span}\\{(2,0),(0,2)\\}\subset\mathbb{Z}^{2}$ (the lattice of
vectors with both even coordinates). One can easily see, that in this case the
quotient group is isomorphic to $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$, so $L$
is not a sublattice. However the restriction $X_{L}$ is still a subshift of
$\mathcal{A}^{\mathbb{Z}^{2}}$ and at least some of the results of [2] hold in
this case. This led us to a hypothesis, that the projectional entropy may be
well defined for a restriction to any normal subgroup. In the current paper we
attempt to transfer the results of [1] and [2] to the case of actions of
amenable groups. The main theorem states that if a certain mixing condition
holds, namely the system is strongly irreducible, the equality of the entropy
and the projectional entropy with respect to a certain normal subgroup implies
that the system has a product-like structure.

##  2\. Basic notions

To simplify the notation, for a countable set $A$ let $\mathcal{F}(A)$ denote
the family of all finite and nonempty subsets of $A$.

######  Definition 2.1.

Let $G$ be a countable group. A sequence $F_{n}\in\mathcal{F}(G)$ is a Følner
sequence, if it satisfies the following condition:

$\forall_{g\in G}~{}~{}\lim_{n\to\infty}\frac{|gF_{n}\bigtriangleup
F_{n}|}{|F_{n}|}=0,$

where $gF=\\{gf:f\in F\\}$ for $g\in G,F\subset G$.

We call $G$ amenable, if there exists a Følner sequence in $G$.

It is obvious, that if $(F_{n})$ is a Følner sequence, then for any nonempty,
finite $T\subset G$ the term $\frac{|TF_{n}\bigtriangleup F_{n}|}{|F_{n}|}$
converges to $0$ (for $A,B\subset G$ by $AB$ we denote the set $\\{ab:a\in
A,b\in B\\}$). Also the following holds.

######  Lemma 2.2.

Let $(F_{n})$ be a Følner sequence in $G$ and let $B$ be a nonempty finite
subset of $G$. Then $\tilde{F}_{n}:=BF_{n}$ is also a Følner sequence in $G$.

###### Proof.

Fix $g\in G$. From triangle inequality for symmetric difference and the fact
that $|BF_{n}|\geqslant|F_{n}|$ we obtain:

$\frac{|g\tilde{F}_{n}\bigtriangleup\tilde{F}_{n}|}{|\tilde{F}_{n}|}\leqslant\frac{|gBF_{n}\bigtriangleup
F_{n}|}{|F_{n}|}+\frac{|F_{n}\bigtriangleup BF_{n}|}{|F_{n}|}.$

Since $B$ (and so $gB$) is nonempty and finite, and $(F_{n})$ is a Følner
sequence, both terms on the right converge to zero. ∎

The fact below is widely known.

######  Fact 2.3.

Let $G$ be a countable group and $H$ be its normal subgroup. Then

$G\textrm{ is amenable }\iff H\textrm{ and }G/H\textrm{ are amenable.}$

######  Definition 2.4.

Let $G$ be a countable amenable group and let $\mathcal{A}$ be a finite set
with the discrete topology. We define the shift action of $G$ on
$\mathcal{A}^{G}$ as follows:

$\forall_{x\in\mathcal{A}^{G}}~{}\forall_{g,h\in G}~{}~{}(gx)(h)=x(hg).$

The full $G$-shift (or just the full shift, if $G$ is known from the context)
is the product space $\mathcal{A}^{G}$ equipped with the product topology and
the shift action of $G$. We will call the set $\mathcal{A}$ the alphabet, and
its elements will be symbols.

Each element of the full $G$-shift is in fact a function $x:G\to\mathcal{A}$
thus it is reasonable to consider its restriction $x|_{F}$ to a set $F\subset
G$. Let us notice that the full $G$-shift is, by the Tikhonov’s theorem, a
compact topological space. It is well known that the the full shift is
metrizable and that a sequence $x_{n}\in\mathcal{A}^{G}$ converges to
$x\in\mathcal{A}^{G}$ if and only if for any $g\in G$ we have $x_{n}(g)=x(g)$
for $n$ bigger than some $N_{g}$.

We say that $Y\subset\mathcal{A}^{G}$ is $G$-invariant if it is invariant
under the shift action of $G$, i.e. $gY=\\{gy:y\in Y\\}=Y$ for any $g\in G$.

######  Definition 2.5.

A nonempty, compact and $G$-invariant set $X\subset\mathcal{A}^{G}$ equipped
with the shift action of $G$ will be called a $G$-subshift or just a subshift
if $G$ is known from the context.

######  Definition 2.6.

Let $F\in\mathcal{F}(G)$. A configuration on $F$ is an element of
$\mathcal{A}^{F}$. We say that a configuration $\mathsf{C}$ on $F$ appears in
$x\in\mathcal{A}^{G}$ if $x|_{F}=\mathsf{C}$.

For a subshift $X$ by $\mathcal{L}_{F}(X)$ we denote the set of all
configurations on $F$ that appear in the elements of $X$. The set
$\mathcal{L}(X)=\bigcup_{F\in\mathcal{F}(G)}\mathcal{L}_{F}(X)$ is called the
language of $X$.

Similarly to the classical case, one obtains an equivalent definition of a
subshift by forbidding configurations.

######  Theorem 2.7.

$X\subset\mathcal{A}^{G}$ is a subshift if and only if there exists
$\mathcal{N}\subset\bigcup_{F\in\mathcal{F}(G)}\mathcal{A}^{F}$ (a set of
configurations) such that

$X=\\{x\in\mathcal{A}^{G}:\forall_{g\in
G}\forall_{\mathsf{C}\in\mathcal{N}}~{}\mathsf{C}\textrm{ does not appear in
}gx\\}.$

We will use the following simple fact.

######  Fact 2.8.

If $Y$ is a proper subshift of $X$ (that is $Y\subsetneq X$), then
$\mathcal{L}(Y)\subsetneq\mathcal{L}(X)$.

Finally, we define the entropy of a $G$-subshift. The theorem below is a
consequence of the Ornstein-Weiss Lemma (see [3], Theorem 6.1).

######  Theorem 2.9.

Let $X$ be a $G$-subshift. For any Følner sequence $(F_{n})$ in $G$ the limit

$\lim_{n\to\infty}\frac{1}{|F_{n}|}\log\left|\mathcal{L}_{F_{n}}(X)\right|$

exists, is finite, and does not depend on the choice of $(F_{n})$. The common
value $h(X)$ of the limit is called the entropy of $X$.

We will use the following result, which was introduced in [4] and is called
the “infimum rule”.

######  Theorem 2.10.

The following equality holds:

$h(X)=\inf_{F\in\mathcal{F}(G)}\frac{1}{|F|}\log\left|\mathcal{L}_{F}(X)\right|.$

##  3\. The results

Throughout this section $G$ is a fixed countable amenable group, $\mathcal{A}$
is a finite alphabet and $X\subset\mathcal{A}^{G}$ is a $G$-subshift.
Additionally, let $H$ denote a fixed normal subgroup of $G$.

######  Definition 3.1.

A projection of $X$ onto $H$ is a subset $X_{H}$ of $\mathcal{A}^{H}$ defined
as follows:

$X_{H}=\left\\{x^{\prime}\in\mathcal{A}^{H}:\exists_{x\in
X}\textrm{~{}}x^{\prime}=x|_{H}\right\\}.$

By Fact 2.3 we know that $H$ is amenable, so we can formulate the following
theorem.

######  Theorem 3.2.

$X_{H}$ is an $H$-subshift.

###### Proof.

$H$-invariance. Let $h\in H$ and $x^{\prime}\in X_{H}$. By the definition,
there exists $x\in X$ such that $x|_{H}=x^{\prime}$. For any $\tilde{h}\in H$
we have:

$hx^{\prime}(\tilde{h})=x^{\prime}(\tilde{h}h)=x(\tilde{h}h)=hx(\tilde{h}),$

hence $hx^{\prime}=(hx)|_{H}$. The subshift $X$ is $G$-invariant, so $hx\in
X$, and thus $hx^{\prime}\in X_{H}$. Therefore we get:

$\forall_{h\in H}~{}~{}hX_{H}\subset X_{H}.$

But that means that for any $h\in H$ also $h^{-1}X_{H}\subset X_{H}$. Hence
$X_{H}=h(h^{-1}X_{H})\subset hX_{H}$, for any $h\in H$.

Compactness. Let $(x^{\prime}_{n})_{n\geqslant 1}$ be a sequence in $X_{H}$.
We know that for any $n$ there exists $x_{n}\in X$ such that
$x^{\prime}_{n}=x_{n}|_{H}$. By the compactness of $X$ there is a subsequence
$x_{n_{k}}$ convergent to some $x\in X$. Then for each $g\in G$ $x_{n_{k}}(g)$
eventually equals $x(g)$, in particular it holds for each $g\in H$. This gives
us the convergence $x^{\prime}_{n_{k}}\rightarrow x|_{H}\in X_{H}$. ∎

The following definition introduces our main object of interest.

######  Definition 3.3.

The entropy of $X_{H}$ (as a subshift of $\mathcal{A}^{H}$) is the
$H$-projectional entropy of $X$.

Below we present a method of constructing a subshift of $\mathcal{A}^{G}$ from
$X_{H}$ in a “product-like” way.

######  Definition 3.4.

A transversal for $H$ is a subset $M$ of $G$ such that for any coset $gH$ we
have $|M\cap gH|=1$ (i.e. $M$ consists of representatives of the cosets).

For a transversal $M$ for $H$ we define a function
$\varphi_{M}:\left(X_{H}\right)^{G/H}\rightarrow\mathcal{A}^{G}$ in the
following way. For any $g\in G$ there exists exactly one $h\in H$ and exactly
one $m\in M$ such that $g=hm$. We take

$\varphi_{M}\left((x^{\gamma})_{\gamma\in G/H}\right)(g)=x^{Hm}(h),$

where $x^{\gamma}\in X_{H}$ for any coset $\gamma\in G/H$.

######  Theorem 3.5.

  1. (1)

For each transversal $M$ the function $\varphi_{M}$ is an injection.

  2. (2)

For any two transversals $M_{1},M_{2}$ we have

$\varphi_{M_{1}}\left(\left(X_{H}\right)^{G/H}\right)=\varphi_{M_{2}}\left(\left(X_{H}\right)^{G/H}\right).$

We will denote the common image by $X_{H}^{G/H}$.

  3. (3)

For each transversal $M$ the function $\varphi_{M}$ is continuous with respect
to the product topology on $\left(X_{H}\right)^{G/H}$.

  4. (4)

$X_{H}^{G/H}$ is a $G$-subshift.

Let us note that by $\left(X_{H}\right)^{G/H}$ we denote the Cartesian power
of the set $X_{H}$, while $X_{H}^{G/H}$ is a subset of $\mathcal{A}^{G}$.

###### Proof.

We will use the notation
$\left(x^{\gamma}\right)=\left(x^{\gamma}\right)_{\gamma\in G/H}$.

  1. (1)

Let $\left(x^{\gamma}\right)\neq\left(y^{\gamma}\right)$. Then there exists
$\gamma_{0}\in G/H$ such that $x^{\gamma_{0}}\neq y^{\gamma_{0}}$, which means
that there exists $h_{0}\in H$ such that $x^{\gamma_{0}}(h_{0})\neq
y^{\gamma_{0}}(h_{0})$. Let $m_{0}\in M\cap\gamma_{0}$ (in fact
$M\cap\gamma_{0}=\\{m_{0}\\}$). Then we have

$\varphi_{M}\left(\left(x^{\gamma}\right)\right)(h_{0}m_{0})=x^{\gamma_{0}}(h_{0})\neq
y^{\gamma_{0}}(h_{0})=\varphi_{M}\left(\left(y^{\gamma}\right)\right)(h_{0}m_{0}),$

so
$\varphi_{M}\left(\left(x^{\gamma}\right)\right)\neq\varphi_{M}\left(\left(y^{\gamma}\right)\right)$.

  2. (2)

Let $M_{1}$, $M_{2}$ be transversals for $H$. By the symmetry it suffices to
show that

$\varphi_{M_{1}}\left(\left(X_{H}\right)^{G/H}\right)\subset\varphi_{M_{2}}\left(\left(X_{H}\right)^{G/H}\right).$

Fix $\left(x^{\gamma}\right)\in\left(X_{H}\right)^{G/H}$. For any $\gamma\in
G/H$ let $m_{i}^{\gamma}$ be the only element of $M_{i}\cap\gamma$, $i=1,2$,
and take
$y^{\gamma}=m_{2}^{\gamma}\left(m_{1}^{\gamma}\right)^{-1}x^{\gamma}\in\mathcal{A}^{H}$.
It is well defined, because $m_{2}^{\gamma}\left(m_{1}^{\gamma}\right)^{-1}\in
H$. We have $y^{\gamma}\in X_{H}$ for any $\gamma$, since $X_{H}$ is
$H$-invariant.

Fix $g\in G$ and take $\gamma_{0}=gH$. There exists exactly one pair
$h_{1},h_{2}\in H$ such that
$g=h_{1}m_{1}^{\gamma_{0}}=h_{2}m_{2}^{\gamma_{0}}$. Note that then
$h_{1}=h_{2}m_{2}^{\gamma_{0}}\left(m_{1}^{\gamma_{0}}\right)^{-1}$. We have

$\varphi_{M_{1}}\left(\left(x^{\gamma}\right)\right)(g)=x^{\gamma_{0}}(h_{1})=x^{\gamma_{0}}\left(h_{2}m_{2}^{\gamma_{0}}\left(m_{1}^{\gamma_{0}}\right)^{-1}\right)$
$=m_{2}^{\gamma_{0}}\left(m_{1}^{\gamma_{0}}\right)^{-1}x^{\gamma_{0}}(h_{2})=y^{\gamma_{0}}(h_{2})=\varphi_{M_{2}}\left(\left(y^{\gamma}\right)\right)(g).$

Since $g$ and $(x^{\gamma})$ were arbitrary, we get that

$\forall_{(x^{\gamma})\in(X_{H})^{G/H}}~{}\exists_{(y^{\gamma})\in(X_{H})^{G/H}}~{}~{}\varphi_{M_{1}}((x^{\gamma}))=\varphi_{M_{2}}((y^{\gamma})),$

thus
$\varphi_{M_{1}}\left(\left(X_{H}\right)^{G/H}\right)\subset\varphi_{M_{2}}\left(\left(X_{H}\right)^{G/H}\right).$

  3. (3)

We skip the proof of continuity, as it boils down to verifying that the
preimage of a cylinder is also a cylinder.

  4. (4)

$G$-invariance: By the same argument as in the proof of Theorem 3.2 it
suffices to show that for any $g\in G$ we have $gX_{H}^{G/H}\subset
X_{H}^{G/H}$.

Fix a transversal $M$. For any $g\in G$ by $h^{g}$ we denote the element of
$H$ such that $g=h^{g}m^{gH}$, where $m^{gH}$ is the only element of $M\cap
gH$.

Fix $g\in G$ and $x=\varphi_{M}\left(\left(x^{\gamma}\right)\right)\in
X_{H}^{G/H}$. For any $\tilde{g}\in G$ we have

$h^{\tilde{g}g}=\tilde{g}g(m^{\tilde{g}gH})^{-1}=h^{\tilde{g}}m^{\tilde{g}H}g(m^{\tilde{g}gH})^{-1}.$

Therefore $m^{\tilde{g}H}g(m^{\tilde{g}gH})^{-1}\in H$ and

$gx(\tilde{g})=x(\tilde{g}g)=x\left(h^{\tilde{g}g}m^{\tilde{g}gH}\right)\\\
=x^{\tilde{g}gH}\left(h^{\tilde{g}g}\right)=x^{\tilde{g}H\cdot
gH}\left(h^{\tilde{g}}m^{\tilde{g}H}g(m^{\tilde{g}gH})^{-1}\right)\\\
=\left(m^{\tilde{g}H}g(m^{\tilde{g}H\cdot gH})^{-1}\right)x^{\tilde{g}H\cdot
gH}\left(h^{\tilde{g}}\right),$

where $\tilde{g}H\cdot gH=\tilde{g}gH$ denotes the product of cosets in the
quotient group $G/H$.

Now, for each $\gamma\in G/H$ define:

$y^{\gamma}:=\left(m^{\gamma}g(m^{\gamma\cdot gH})^{-1}\right)x^{\gamma\cdot
gH}\in X_{H}.$

For any $\tilde{g}\in G$ we have

$\varphi_{M}\left(\left(y^{\gamma}\right)\right)(\tilde{g})=\left(m^{\tilde{g}H}g\left(m^{\tilde{g}H\cdot
gH}\right)^{-1}\right)x^{\tilde{g}H\cdot gH}(h^{\tilde{g}})=gx(\tilde{g}).$

Hence we have $gx\in X_{H}^{G/H}$. Since $x$ was arbitrary,
$gX_{H}^{G/H}\subset X_{H}^{G/H}$.

Compactness is clear, because $X_{H}^{G/H}$ is an image of the compact set
$(X_{H})^{G/H}$ under the continuous function $\varphi_{M}$.

∎

From now on $M$ is a fixed transversal for $H$ and $\varphi=\varphi_{M}$

######  Fact 3.6.

$X\subset X_{H}^{G/H}$

###### Proof.

Take any $x\in X$ and define $x^{Hm}:=(mx)|_{H}$ for any $m\in M$. Then
$x=\varphi\left(\left(x^{\gamma}\right)\right)$ (we have
$x^{Hm}(h)=mx(h)=x(hm)$). ∎

######  Lemma 3.7.

Let $F_{1},\ldots,F_{n}\in\mathcal{F}(G)$ be subsets of different cosets. Then
for $F=\bigcupdot_{i=1}^{n}F_{i}$ we have

$\mathcal{L}_{F}(X_{H}^{G/H})=\\{\mathsf{C}\in\mathcal{A}^{F}:\forall_{i=1,\ldots,n}~{}\mathsf{C}|_{F_{i}}\in\mathcal{L}_{F_{i}}(X_{H}^{G/H})\\},$

and hence

$\left|\mathcal{L}_{F}(X_{H}^{G/H})\right|=\prod_{i=1}^{n}\left|\mathcal{L}_{F_{i}}(X_{H}^{G/H})\right|.$

###### Proof.

It is obvious that the set on the right hand side contains
$\mathcal{L}_{F}(X_{H}^{G/H})$. To see the opposite inclusion, take any
configurations $\mathsf{C}_{i}\in\mathcal{L}_{F_{i}}(X_{H}^{G/H})$,
$i=1,\ldots,n$, and let configuration $\mathsf{C}$ on $F$ be such that for any
$i$ we have $\mathsf{C}|_{F_{i}}=\mathsf{C}_{i}$. For any $i$ there exists
$x_{i}=\varphi((x_{i}^{\gamma}))\in X_{H}^{G/H}$ such that
$x_{i}|_{F_{i}}=\mathsf{C}_{i}$. Let $m_{i}\in M$ be such that $F_{i}\subset
Hm_{i}$. By the definition there exists $x\in X_{H}^{G/H}$ such that for any
$i$ we have $(m_{i}x)|_{H}=x_{i}^{Hm_{i}}$, which is equivalent to
$x|_{Hm_{i}}=x_{i}|_{Hm_{i}}$. Hence we get that

$\forall_{i=1,\ldots,n}~{}~{}x|_{F_{i}}=x_{i}|_{F_{i}}=\mathsf{C}_{i},$

so $\mathsf{C}=x|_{F}\in\mathcal{L}_{F}(X_{H}^{G/H})$. ∎

The proof of the following theorem was suggested by Tomasz Downarowicz.

######  Theorem 3.8.

$h(X_{H}^{G/H})=h(X_{H})$

###### Proof.

First we prove that $h(X_{H}^{G/H})\leqslant h(X_{H})$. By using Theorem 2.10
we get

$h(X_{H}^{G/H})=\inf_{F\in\mathcal{F}(G)}\frac{1}{|F|}\log\left|\mathcal{L}_{F}(X_{H}^{G/H})\right|\leqslant\inf_{F\in\mathcal{F}(H)}\frac{1}{|F|}\log\left|\mathcal{L}_{F}(X_{H}^{G/H})\right|\\\
=\inf_{F\in\mathcal{F}(H)}\frac{1}{|F|}\log\left|\mathcal{L}_{F}(X_{H})\right|=h(X_{H}).$

To prove $h(X_{H}^{G/H})\geqslant h(X_{H})$, take any $F\in\mathcal{F}(G)$.
Let $m_{1},\ldots,m_{n}$ be all elements of $M$ such that $F_{i}:=F\cap
Hm_{i}\neq\varnothing$ for $i=1,\ldots,n$. It is obvious that
$F=\bigcupdot_{i=1}^{n}F_{i}$. By the preceding lemma and the definition of
$X_{H}^{G/H}$ we have

$\left|\mathcal{L}_{F}(X_{H}^{G/H})\right|=\prod_{i=1}^{n}\left|\mathcal{L}_{F_{i}}(X_{H}^{G/H})\right|=\prod_{i=1}^{n}\left|\mathcal{L}_{F_{i}m_{i}^{-1}}(X_{H})\right|,$

so by using Theorem 2.10

$\frac{1}{|F|}\log\left|\mathcal{L}_{F}(X_{H}^{G/H})\right|=\frac{1}{|F|}\sum_{i=1}^{n}\log\left|\mathcal{L}_{F_{i}m_{i}^{-1}}(X_{H})\right|\\\
=\frac{1}{\sum_{i=1}^{n}|F_{i}|}\sum_{i=1}^{n}|F_{i}|\frac{1}{|F_{i}m_{i}^{-1}|}\log\left|\mathcal{L}_{F_{i}m_{i}^{-1}}(X_{H})\right|\\\
\geqslant\frac{1}{\sum_{i=1}^{n}|F_{i}|}\sum_{i=1}^{n}|F_{i}|h(X_{H})=h(X_{H}).$

∎

Now we turn to the main question—when does the equality of the entropies of
$X$ and $X_{H}$ imply $X=X_{H}^{G/H}$? In [1] and [2] the counterexamples are
given that such implication fails in general. As it was done in these papers,
in order to obtain a positive result we additionally assume a proper mixing
condition.

######  Definition 3.9.

Let $D\in\mathcal{F}(G)$ contain the neutral element of $G$. We say that $X$
is $D$-strongly irreducible, if for any finite $B_{1},B_{2}\subset G$ such
that $DB_{1}\cap B_{2}=\varnothing$ we have

$\forall_{x,y\in X}\exists_{z\in X}~{}~{}z|_{B_{1}}=x|_{B_{1}}\land
z|_{B_{2}}=y|_{B_{2}}.$

We remark that in case of multidimensional subshifts this condition is
stronger than the uniform filling property, which was sufficient for results
of [2].

From now on $D\in\mathcal{F}(G)$ is fixed.

In the course of proving the main theorem we will exploit the notion of the
lower Banach density. We briefly recall its definition and formulate an
auxiliary lemma.

######  Definition 3.10.

Let $A\subset G$. The lower Banach density of $A$ is a number

$\underline{{d}}(A):=\sup_{F\in\mathcal{F}(G)}~{}\inf_{g\in G}\frac{|A\cap
Fg|}{|F|}.$

It is known that $\underline{d}(A)=\lim_{n\to\infty}\inf_{g\in G}\frac{|A\cap
F_{n}g|}{|F_{n}|}$, where $(F_{n})$ is an arbitrary Følner sequence in $G$.

######  Lemma 3.11.

For any finite $B\subset G$ there exists $P\subset G$ such that:

  1. (1)

$P$ has positive lower Banach density,

  2. (2)

for any $g,\tilde{g}\in P$, if $g\neq\tilde{g}$, then $Bg\cap
B\tilde{g}=\varnothing.$

For the proof see e.g. [5] (Corollary 1.11).

To prove the next theorem we adapt the ideas used in [2].

######  Theorem 3.12.

Let $X$ be $D$-strongly irreducible. Then for any proper subshift $Y\subsetneq
X$ we have $h(Y)<h(X).$

###### Proof.

Since $Y$ is a proper subshift, there exists, by Fact 2.8, a configuration
$\mathsf{C}\in\mathcal{L}(X)\setminus\mathcal{L}(Y)$ on some
$B\in\mathcal{F}(G)$. Take $\tilde{B}:=DB$. From Lemma 3.11 there exists
$P\subset G$ such that $\underline{d}(P)>0$ and for any different
$g,\tilde{g}\in P$ we have $\tilde{B}g\cap\tilde{B}\tilde{g}=\varnothing$.

Let $(F_{n})$ be a Følner sequence in $G$. We know that

$\underline{d}(P)=\lim_{n\rightarrow\infty}\inf_{g\in G}\frac{|P\cap
F_{n}g|}{|F_{n}|}.$

Therefore for $n$ greater than some $N$ we have

$\frac{|P\cap F_{n}|}{|F_{n}|}\geqslant\inf_{g\in G}\frac{|P\cap
F_{n}g|}{|F_{n}|}\geqslant\underline{d}(P)-\frac{\underline{d}(P)}{2}=\frac{\underline{d}(P)}{2}>0.$

Hence for $n>N$ we have

(3.1) $|P\cap F_{n}|\geqslant\frac{\underline{d}(P)}{2}|F_{n}|.$

Now, define $\tilde{F}_{n}:=\tilde{B}F_{n}$. By Lemma 2.2 $(\tilde{F}_{n})$ is
a Følner sequence in $G$. Fix $n>N$ and take $J:=P\cap F_{n}$ (notice, that
for any $g\in J$ we have $\tilde{B}g\subset\tilde{F}_{n}$). For any $I\subset
J$ we define

$\mathcal{L}_{n}^{I}:=\left\\{x|_{\tilde{F}_{n}}:x\in X\land\forall_{g\in
J\setminus I}\textrm{~{}}(gx)|_{B}\neq\mathsf{C}\right\\}.$

In particular, $\mathcal{L}_{n}^{J}=\mathcal{L}_{\tilde{F}_{n}}(X)$ and
$\mathcal{L}_{n}^{\varnothing}=\left\\{x|_{\tilde{F}_{n}}:x\in
X\land\forall_{g\in J}\textrm{~{}}(gx)|_{B}\neq\mathsf{C}\right\\}$.

Let us enumerate elements of $J$: $J=\\{g_{1}\textrm{, },g_{2}\textrm{,
},\ldots,g_{|J|}\\}$. For any $I=\\{g_{1},\ldots,g_{i}\\}$, where $i<|J|$, we
have

$\left|\mathcal{L}_{n}^{I}\right|=\left|\mathcal{L}_{n}^{I\cup\\{g_{i+1}\\}}\setminus\left\\{x|_{\tilde{F}_{n}}:x\in
X\land
x|_{\tilde{F}_{n}}\in\mathcal{L}_{n}^{I\cup\\{g_{i+1}\\}}\land(g_{i+1}x)|_{B}=\mathsf{C}\right\\}\right|$
(3.2)
$=\left|\mathcal{L}_{n}^{I\cup\\{g_{i+1}\\}}\right|-\left|\left\\{x|_{\tilde{F}_{n}}:x\in
X\land
x|_{\tilde{F}_{n}}\in\mathcal{L}_{n}^{I\cup\\{g_{i+1}\\}}\land(g_{i+1}x)|_{B}=\mathsf{C}\right\\}\right|$

(we allow $I$ to be empty, then we take $i=0$).

On the other hand, for any nonempty $I\subset J$ and $g\in I$ we have

$\left|\mathcal{L}_{n}^{I}\right|\leqslant\left|\mathcal{L}_{\tilde{B}g}(X)\right|\left|\left\\{x|_{\tilde{F}_{n}\setminus\tilde{B}g}:x\in
X\land x|_{\tilde{F}_{n}}\in\mathcal{L}_{n}^{I}\right\\}\right|$
$\leqslant\left|\mathcal{L}_{\tilde{B}}(X)\right|\left|\left\\{x|_{\tilde{F}_{n}}:x\in
X\land
x|_{\tilde{F}_{n}}\in\mathcal{L}_{n}^{I}\land(gx)|_{B}=\mathsf{C}\right\\}\right|,$

where the second inequality comes from the facts that $X$ is $D$-strongly
irreducible and that
$\left|\mathcal{L}_{\tilde{B}g}(X)\right|=\left|\mathcal{L}_{\tilde{B}}(X)\right|.$
Hence for any nonempty $I\subset J$ and any $g\in I$ we get

$\left|\left\\{x|_{\tilde{F}_{n}}:x\in X\land
x|_{\tilde{F}_{n}}\in\mathcal{L}_{n}^{I}\land
x|_{Bg}=\mathsf{C}\right\\}\right|\geqslant\left|\mathcal{L}_{\tilde{B}}(X)\right|^{-1}\left|\mathcal{L}_{n}^{I}\right|.$

From (3.2) and the above we obtain

(3.3) $\forall_{I=\\{g_{1},\ldots,g_{i}\\}\textrm{,
}i<|J|}~{}~{}\left|\mathcal{L}_{n}^{I}\right|\leqslant\left|\mathcal{L}_{n}^{I\cup\\{g_{i+1}\\}}\right|\left(1-\left|L_{\tilde{B}}(X)\right|^{-1}\right).$

Now, let us notice that
$\mathcal{L}_{\tilde{F}_{n}}(Y)\subset\mathcal{L}_{n}^{\varnothing}$.
Therefore, by using (3.3)

$\left|\mathcal{L}_{\tilde{F}_{n}}(Y)\right|\leqslant\left|\mathcal{L}_{n}^{\varnothing}\right|\leqslant\left|\mathcal{L}_{n}^{\\{g_{1}\\}}\right|\left(1-\left|L_{\tilde{B}}(X)\right|^{-1}\right).$

Inductively, by using (3.3) $\left(|J|-1\right)$ times more, we get

$\left|\mathcal{L}_{\tilde{F}_{n}}(Y)\right|\leqslant\left|\mathcal{L}_{n}^{J}\right|\left(1-\left|L_{\tilde{B}}(X)\right|^{-1}\right)^{|J|}$
$\leqslant\left|\mathcal{L}_{n}^{J}\right|\left(1-\left|L_{\tilde{B}}(X)\right|^{-1}\right)^{\frac{\underline{d}(P)}{2}|F_{n}|}.$

The last inequality comes from (3.1) and the fact that
$\left(1-\left|L_{\tilde{B}}(X)\right|^{-1}\right)\in(0,1)$. Hence we get

$\frac{1}{\left|\tilde{F}_{n}\right|}\log{\left|\mathcal{L}_{\tilde{F}_{n}}(Y)\right|}\leqslant\frac{1}{\left|\tilde{F}_{n}\right|}\log{\left|\mathcal{L}_{n}^{J}\right|}+\frac{\frac{\underline{d}(P)}{2}|F_{n}|}{\left|\tilde{B}F_{n}\right|}\log{\left(1-\left|L_{\tilde{B}}(X)\right|^{-1}\right)}\\\
\leqslant\frac{1}{\left|\tilde{F}_{n}\right|}\log{\left|\mathcal{L}_{n}^{J}\right|}+\frac{\frac{\underline{d}(P)}{2}|F_{n}|}{\left|\tilde{B}\right|\left|F_{n}\right|}\log{\left(1-\left|L_{\tilde{B}}(X)\right|^{-1}\right)}\\\
=\frac{1}{\left|\tilde{F}_{n}\right|}\log{\left|\mathcal{L}_{\tilde{F}_{n}}(X)\right|}+\frac{\frac{\underline{d}(P)}{2}}{\left|\tilde{B}\right|}\log{\left(1-\left|L_{\tilde{B}}(X)\right|^{-1}\right)}.$

By letting $n$ go to infinity ($n$ was fixed, greater than $N$) we obtain

$h(Y)\leqslant
h(X)+\frac{\underline{d}(P)}{2\left|\tilde{B}\right|}\log{\left(1-\left|L_{\tilde{B}}(X)\right|^{-1}\right)}<h(X).$

∎

######  Theorem 3.13.

If $X$ is $D$-strongly irreducible, then so is $X_{H}^{G/H}$.

###### Proof.

Let $B_{1},B_{2}\subset G$ be finite and such that $DB_{1}\cap
B_{2}=\varnothing$. Take any $x,y\in X_{H}^{G/H}$ and let
$(x^{\gamma}),(y^{\gamma})\in(X_{H})^{G/H}$ be such that
$x=\varphi((x^{\gamma}))$ and $y=\varphi((y^{\gamma}))$. Write
$M=\\{m_{1},m_{2},\ldots\\}$. For any $i=1,2,\ldots$ define $S_{i}=B_{1}\cap
Hm_{i}$ and $T_{i}=B_{2}\cap Hm_{i}$.

By the definition of $\varphi$ for each $i$ we have
$(m_{i}x)|_{S_{i}m_{i}^{-1}}=x^{Hm_{i}}|_{S_{i}m_{i}^{-1}}$ and
$(m_{i}y)|_{T_{i}m_{i}^{-1}}=y^{Hm_{i}}|_{T_{i}m_{i}^{-1}}$ (since $S_{i}$ and
$T_{i}$ are subsets of $Hm_{i}$ and $H$ is normal, $S_{i}m_{i}^{-1}$ and
$T_{i}m_{i}^{-1}$ are subsets of $H$). Moreover, there exist $x_{i},y_{i}\in
X$ such that $x^{Hm_{i}}=x_{i}|_{H}$ and $y^{Hm_{i}}=y_{i}|_{H}$. Since $X$ is
$D$-strongly irreducible and

$DS_{i}m_{i}^{-1}\cap T_{i}m_{i}^{-1}\subset(DB_{1}\cap
B_{2})m_{i}^{-1}=\varnothing,$

there exists $z_{i}\in X$ such that
$z_{i}|_{S_{i}m_{i}^{-1}}=x_{i}|_{S_{i}m_{i}^{-1}}$ and
$z_{i}|_{T_{i}m_{i}^{-1}}=y_{i}|_{T_{i}m_{i}^{-1}}$. By taking
$z^{Hm_{i}}=z_{i}|_{H}$ for any $i$ and defining $z=\varphi((z^{\gamma}))$ we
get $z|_{S_{i}}=x|_{S_{i}}$ and $z|_{T_{i}}=y|_{T_{i}}$ for any $i$. Therefore
$z|_{B_{1}}=|x_{B_{1}}$ and $z|_{B_{2}}=|y_{B_{2}}$. ∎

We are now ready to formulate the main theorem.

######  Theorem 3.14.

If $X$ is $D$-strongly irreducible and $h(X)=h(X_{H})$, then $X=X_{H}^{G/H}$.

###### Proof.

From Fact 3.6 we know that $X\subset X_{H}^{G/H}$. Assume that $X\subsetneq
X_{H}^{G/H}$. By Theorem 3.13 $X_{H}^{G/H}$ is $D$-strongly irreducible. Then
by theorems 3.12 and 3.8 we get $h(X)<h(X_{H})$. ∎

Aknowledgements. First of all I want to thank my supervisor Bartosz Frej, who
introduced me into the subject and was helping me during my work. I am also
grateful to Tomasz Downarowicz for lots of useful advice.

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