# A Piecewise Smooth Fermi-Ulam Pingpong with Potential

Jing Zhou  Department of Mathematics, University of Maryland, College Park, MD
20742<EMAIL_ADDRESS>
###### Abstract.

In this paper we study a Fermi-Ulam model where a pingpong bounces elastically
against a periodically oscillating platform in a gravity field. We assume that
the platform motion $f(t)$ is piecewise $C^{3}$ with a singularity
$\dot{f}(0+)\neq\dot{f}(1-)$. If the second derivative of the platform motion
is either always positive $\ddot{f}(t)>0$ or always $\ddot{f}(t)<-g$ where $g$
is the gravitational constant, then the escaping orbits constitute a null set
and the system is recurrent. However, under these assumptions, escaping orbits
coexist with bounded orbits at arbitrarily high energy level.

The author is much obliged to Dmitry Dolgopyat for proposing the problem and
generously providing advice. The author also wants to thank Hong-Kun Zhang for
her help and suggestions. This research was partially supported by the Patrick
and Marguerite Sung Fellowship in Mathematics at University of Maryland and
the NSF grant DMS 1665046.

##  1\. Introduction

There has been an extensive study on the Fermi-Ulam pingpong models since
Fermi [15] and Ulam [28] proposed the bouncing ball mechanism as an
explanation for the existence of high energy particles in the cosmic rays. The
original Fermi-Ulam model describes a point particle bouncing elastically
between two infinitely heavy wall, one fixed and the other oscillating
periodically [28]. Ulam conjectured [28], based on his numerical experiment
with piecewise linearly oscillating wall, the existence of escaping orbits,
i.e. orbits whose energy grows to infinity in time. In addition, bounded
orbits (i.e. those whose energy always stays bounded) and oscillatory obits
(i.e. those whose energy has a finite liminf but infinite limsup) might also
exist in Fermi-Ulam models and various attempts have been made to examine the
existence and prevalence of each of these three types of orbits (the author
refers to [12, 16, 19] for surveys).
Later KAM theory has negated the existence of accelerating orbits with
sufficiently smooth wall motions as the prevalence of invariant curves
prevents energy diffusion [17, 25, 26]. In nonsmooth cases, Zharnitsky [29]
found linearly escaping orbits in a piecewise linear model. In a piecewise
smooth model with one singularity de Simoi and Dolgopyat [8] showed that there
exists a parameter determining whether the linear part of the limiting system
at infinity is elliptic or hyperbolic (i.e. whether the absolute value of
trace is less or greater than 2), and that bounded orbits co-exist with
escaping ones in elliptic regimes while escaping orbits have zero measure but
full Hausdorf dimension in hyperbolic regimes.
When background potential is introduced, Arnold and Zharnitsky [2] found
unbounded orbits in a system with switching potentials. If the fixed wall is
removed and gravity is present, Pustylnikov [24] showed that there exists an
open set of wall motions in the space of analytic periodic functions admitting
analytic extension to a fixed strip $|\Im t|<\varepsilon$ which produce
infinite measure of escaping orbits. In a Duffing equation with a polynomial
potential which possesses one discontinuity, Levi and You [18] proved the
existence of oscillatory orbits. Ortega provided conditions for the existence
of escaping orbits in piecewise linear oscillators [21, 22]. For intermediate
cases where the potential takes the form $U=x^{\alpha}$ and the wall motion is
sin-type, Dolgopyat [11] proved that the escaping orbits do not exist for
$\alpha>1,\alpha\neq 2$ and constitute a null set for $\alpha<1/3$, while de
Simoi [7] showed in the same setting that the escaping orbits possess full
Hausdorff dimension for $\alpha<1$.

In this paper we study a Fermi-Ulam pingpong model with a potential. The model
describes that a point mass is bouncing elastically against a moving wall in
the gravity field. The motion (height) of the wall is a piecewise smooth
periodic function $f(t)$ and the gravitational constant is given by $g$.

g Figure 1. Bouncing Pingpong in Gravity Field

We are interested in the case when the motion $f(t)$ of the wall is 1-periodic
and piecewise $C^{3}$, i.e $f(t+1)=f(t)$, $f\in C^{3}(0,1)$ and
$\dot{f}(0+)\neq\dot{f}(1-)$. We record the time $t$ of each collision and the
velocity $v$ immediately after each collision. We exclude from our discussion
the singular collisions at integer times, which form a null set in the
$(t,v)$-phase cylinder. We investigate the dynamics of the model by looking at
the collision map $F$, which sends one collision $(t,v)$ to the next one
$(\bar{t},\bar{v})$. Our main result is that if the second derivative of the
wall motion behaves, i.e. the second derivative is either always positive
$\ddot{f}(t)>0$ or always less than the negative of the gravitational constant
$\ddot{f}(t)<-g$, then the escaping orbits have zero measure and $F$ is
recurrent. We also show that under these assumptions, escaping and bounded
orbits exist at arbitrarily high energy level.

##  2\. Main Results

In this section we state the main results of this paper.

We denote the second derivative of the wall motion as $k(t)=\ddot{f}(t)$. The
collision map $F$ preserves an absolutely continuous measure $\mu=wdtdv$ where
$w=v-\dot{f}$ is the relative velocity after collision (c.f. Section 3.1).

For large velocities, the dynamics can be approximated by

$F(t,v)=F_{\infty}(t,v)+\mathcal{O}\left(\frac{1}{v}\right)$

where

$F_{\infty}(t,v)=\left(t+\frac{2v}{g},v+2\dot{f}\left(t+\frac{2v}{g}\right)\right).$

It is easy to verify that the limit map $F_{\infty}$ is area-preserving and it
covers a map $\tilde{F}_{\infty}$ on the torus
$\mathbb{R}/\mathbb{Z}\times\mathbb{R}/g\mathbb{Z}$

$\tilde{F}_{\infty}:\begin{cases}\tilde{t}_{1}=\tilde{t}_{0}+\frac{2\tilde{v}_{0}}{g}\\\
\tilde{v}_{1}=\tilde{v}_{0}+2\dot{f}(\tilde{t}_{1})\end{cases}$

where $\tilde{t}=t\mbox{ (mod 1)}$, $\tilde{v}=v\mbox{ (mod g})$.

If the second derivative $\ddot{f}$ of the wall motion is either always
positive or always less than $-g$, then the limit map $\tilde{F}_{\infty}$ is
ergodic:

######  Theorem 1.

Suppose that $\ddot{f}(t)>0$ for any $t>0$. Then the map $\tilde{F}_{\infty}$
is ergodic.

######  Theorem 2.

Suppose that for any $t>0$, $\ddot{f}(t)<-g$ where $g$ is the gravitational
constant. Then the map $\tilde{F}_{\infty}$ is ergodic.

We shall call such wall motions _admissible_ if either for all $t$,
$\ddot{f}>0$ or for all $t$, $\ddot{f}<-g$.

######  Remark 2.1.

The map $\tilde{F}_{\infty}$ might not be ergodic if the assumptions in
Theorem 1 and Theorem 2 fail. For example, when the wall motion is analytic
Pustylnikov [24] found an KAM island for the limit map for an open set of
analytic periodic wall motions which admit analytic extension to a strip $|\Im
t|<\varepsilon$. Note that for analytic motions $\int_{0}^{1}\ddot{f}(t)dt=0$
so analytic motions are not admissible.

Besides the above ergodic properties, we also obtain stronger statistical
properties of $\tilde{F}_{\infty}$ under the same assumptions.

For every $x,y\in\mathbb{T}$, we define their forward separation time
$s_{+}(x,y)$ to be the smallest nonnegative integer $n$ such that $x,y$
belongs to distinct continuity components of $\tilde{F}_{\infty}^{n}$. We can
define similarly their backward separation time $s_{-}(x,y)$ for the inverse
iterates.
A function $\varphi:\mathbb{T}\to\mathbb{R}$ is said to be dynamically Hölder
continuous if there exist $\vartheta=\vartheta(\tilde{F}_{\infty})\leq 1$ such
that

$|\varphi|_{\vartheta}^{+}:=\sup\left\\{\frac{|\varphi(x)-\varphi(y)|}{\vartheta^{s_{+}(x,y)}}:\hbox{$x\neq
y$ on the same unstable manifold}\right\\}<\infty,$

and that

$|\varphi|_{\vartheta}^{-}:=\sup\left\\{\frac{\varphi(x)-\varphi(y)|}{\vartheta^{s_{-}(x,y)}}:\hbox{$x\neq
y$ on the same stable manifold}\right\\}<\infty.$

######  Theorem 3 (Exponential Decay of Correlations).

Suppose that the wall motion is admissible. Then the map $\tilde{F}_{\infty}$
enjoys exponential decay of correlations for dynamically Hölder continuous
observables: $\exists\ b>0$ such that for any pair of dynamically Hölder
continuous observables $\varphi,\phi$, $\exists\ C_{\varphi,\phi}$ such that

$\bigg{|}\int_{\mathbb{T}}(\varphi\circ\tilde{F}_{\infty}^{n})\phi
d\tilde{\mu}-\int_{\mathbb{T}}\varphi d\tilde{\mu}\int_{\mathbb{T}}\phi
d\tilde{\mu}\bigg{|}\leq C_{\varphi,\phi}e^{-bn},\ \ n\in\mathbb{N}.$

We observe that for any dynamically Hölder observable $\varphi$, the following
quantity is finite due to Theorem 3

$\sigma_{\varphi}^{2}:=\sum_{n=-\infty}^{\infty}\int_{\mathbb{T}}\varphi\cdot(\varphi\circ\tilde{F}_{\infty}^{n})d\tilde{\mu}<\infty.$

######  Theorem 4 (CLT).

Suppose that the wall motion is admissible. Then the map $\tilde{F}_{\infty}$
satisfies central limit theorem for dynamically Hölder observables, i.e.

$\frac{1}{\sqrt{n}}\sum_{i=0}^{n-1}\varphi\circ\tilde{F}_{\infty}^{i}\stackrel{{\scriptstyle
dist}}{{\rightharpoonup}}\mathcal{N}(0,\sigma_{\varphi}^{2})$

where $\varphi$ is dynamically Hölder with zero average
$\int_{\mathbb{T}}\varphi d\tilde{\mu}=0$.

As for the original system, under the assumptions in Theorem 1 or Theorem 2
the escaping orbit of the collision map $F$ constitute a null set:

######  Theorem 5 (Null Escaping Set).

Suppose that the wall motion is admissible. Then the set $E$ of escaping
orbits of $F$ has zero measure.

It turns out that the escaping set is exactly the dissipative part of the
system and consequently under the assumptions in Theorem 1 or 2 the system is
recurrent:

######  Corollary 6 (Recurrence).

Suppose that the wall motion is admissible. Then $F$ is recurrent, i.e. almost
every orbit comes arbitrarily close to its initial point.

However, under the assumptions in Theorem 1 or Theorem 2, escaping and bounded
orbits still exist:

######  Theorem 7.

Suppose that $f(t)$ is admissible. Then $F$ possesses escaping and bounded
orbits with arbitrarily high energy.

Moreover, $F$ satisfies the following _global global mixing_ property for
_global functions_. We say a function $\Phi$ is _global_ if it is bounded,
uniformly continuous and has a finite average $\bar{\Phi}$ in the following
sense: for any $\epsilon$ there exists $N$ so large that for each square $V$
of size $\mu(V)>N$ we have

$\left|\frac{1}{\mu(V)}\int_{V}\Phi d\mu-\bar{\Phi}\right|\leq\epsilon.$

We denote by $\mathbb{G}_{U}$ the space of all such global functions.

######  Theorem 8 (Global Global Mixing).

Suppose that the wall motion is admissible. Then $F$ is global global mixing
with respect to $\mathbb{G}_{U}$, i.e. for any
$\Phi_{1},\Phi_{2}\in\mathbb{G}_{U}$, the following holds

$\displaystyle\lim_{n\to\infty}\limsup_{\mu(V)\to\infty}\frac{1}{\mu(V)}\int_{V}\Phi_{1}\cdot(\Phi_{2}\circ
F^{n})d\mu$ $\displaystyle=$
$\displaystyle\lim_{n\to\infty}\liminf_{\mu(V)\to\infty}\frac{1}{\mu(V)}\int_{V}\Phi_{1}\cdot(\Phi_{2}\circ
F^{n})d\mu$ $\displaystyle=\bar{\Phi}_{1}\bar{\Phi}_{2}.$

##  3\. Preliminaries

In this section, we discuss the collision map. The study of the collision map
relies substantially on the behavior of its limiting map, i.e. the
approximated collision map for large velocities. We also discuss the
singularity lines/curves of the limit map as they will play a very important
role in the proofs later.

###  3.1. The Collision Map

We denote $s_{n}=t_{n+1}-t_{n}$ the flight time between two consecutive
collisions.

Two consecutive collisions satisfy the following equations

(1) $\left\\{\begin{aligned}
&-(v_{n}-gs_{n}-\dot{f}(t_{n+1}))=v_{n+1}-\dot{f}(t_{n+1})\\\
&f(t_{n})+v_{n}s_{n}-\frac{1}{2}gs_{n}^{2}=f(t_{n+1}).\end{aligned}\right.$

We compute the derivative of the collision map $F$ by differentiating these
equations eq. 1

$dF=\begin{pmatrix}1+\frac{\dot{f}(t_{n})-\dot{f}(t_{n+1})}{w_{n+1}}&\frac{s_{n}}{w_{n+1}}\\\
2\ddot{f}(t_{n+1})+(2\ddot{f}(t_{n+1})+g)\frac{\dot{f}(t_{n})-\dot{f}(t_{n+1})}{w_{n+1}}&(2\ddot{f}(t_{n+1})+g)\frac{s_{n}}{w_{n+1}}-1\end{pmatrix}.$

We observe that $\det dF=\frac{w_{n}}{w_{n+1}}$ and hence $F$ preserves the
measure $\mu=wdtdv$ on the phase cylinder.

###  3.2. The Limit Map

If we only consider collisions with large velocities, the dynamics can be
approximated by

$F(t,v)=F_{\infty}(t,v)+\mathcal{O}\left(\frac{1}{v}\right)$

where

$F_{\infty}(t,v)=\left(t+\frac{2v}{g},v+2\dot{f}\left(t+\frac{2v}{g}\right)\right).$

Denote as $(t_{1},v_{1})=F_{\infty}(t_{0},v_{0})$, then

$t_{1}=t_{0}+\frac{2v_{0}}{g},\quad v_{1}=v_{0}+2\dot{f}(t_{1}).$

As mentioned in Section 2, the limit map $F_{\infty}$ covers a map
$\tilde{F}_{\infty}$ on the torus
$\mathbb{T}=\mathbb{R}/\mathbb{Z}\times\mathbb{R}/g\mathbb{Z}$

$\tilde{t}_{1}=\tilde{t}_{0}+\frac{2\tilde{v}_{0}}{g},\
\tilde{v}_{1}=\tilde{v}_{0}+2\dot{f}(\tilde{t}_{1})$

where $\tilde{t}=t\mbox{ (mod 1)}$, $\tilde{v}=v\mbox{ (mod g})$.

Denote as $k(t)=\ddot{f}(t)$. The dynamics of $\tilde{F}_{\infty}$ can be
decomposed as

$\tilde{t}_{1}=\tilde{t}_{0}+\frac{2\tilde{v}_{0}}{g}\mbox{ (mod 1)},\
\tilde{v}_{0}=\tilde{v}_{0},$

and

$\tilde{t}_{1}=\tilde{t}_{1},\
\tilde{v}_{1}=\tilde{v}_{0}+2\dot{f}(\tilde{t}_{1})$

hence the derivative of $\tilde{F}_{\infty}$ at
$(\tilde{t}_{0},\tilde{v}_{0})$ is

$d_{(\tilde{t}_{0},\tilde{v}_{0})}\tilde{F}_{\infty}=\begin{pmatrix}1&\frac{2}{g}\\\
2k_{1}&\frac{4k_{1}}{g}+1\end{pmatrix}.$

We observe that $\det d\tilde{F}_{\infty}=1$, so $\tilde{F}_{\infty}$
preserves the Lebesgue measure $\tilde{\mu}=d\tilde{t}d\tilde{v}$ on the
torus.

###  3.3. The Singularity Lines of the Limit Map

A singularity occurs when the ball collides with wall at the singularities of
the wall motion, i.e. $t\in\mathbb{N}$, hence the singularity line
$\mathcal{S}^{+}$ of $\tilde{F}_{\infty}$ consists of the points whose next
collisions happen at integer times, i.e.

$\mathcal{S}^{+}=\\{\tilde{t}_{1}=0\\}=\left\\{\tilde{t}_{0}+\frac{2\tilde{v}_{0}}{g}\equiv_{1}0\right\\}.$

Similarly, the singularity line $\mathcal{S}^{-}$ of $\tilde{F}_{\infty}^{-1}$
consists of the points whose the preimages land on integer times, i.e.

$\mathcal{S}^{-}=\\{\tilde{t}_{-1}=0\\}=\left\\{\tilde{t}_{0}+\frac{4}{g}\dot{f}(\tilde{t}_{0})-\frac{2\tilde{v}_{0}}{g}\equiv_{1}0\right\\}.$

We observe that $\mathcal{S}^{\pm}$ consists of finitely many line/curve
segments.

##  4\. Ergodicity of the Limit Map

In this section we establish the ergodicity of the limit map
$\tilde{F}_{\infty}$ under the assumptions in Theorem 1 and Theorem 2. We use
the result by Liverani and Wojtkowski in [20] where they proved ergodicity for
a large class of Hamiltonian systems with invariant cones. We first describe
the class of symplectic maps considered in [20] and then prove that
$\tilde{F}_{\infty}$ satisfies the conditions of [20].

Those conditions involve strictly invariant cones and the least coefficient of
expansion, which are defined as follows.
For a point $p\in X$, we say $p$ possess _strictly monotone cones_ if there
exist cone fields $\mathcal{C}(p)$ and its complementary
$\mathcal{C}^{\prime}(p)$ such that $d_{p}T$ preserves strictly the cone
$\mathcal{C}(p)$ and that $d_{p}T^{-1}$ preserves strictly its complementary
cone $\mathcal{C}^{\prime}(p)$.
For $p\in X$ with strictly monotone cones
$\mathcal{C}(p),\mathcal{C}^{\prime}(p)$, there is an associated defining
quadratic form $\mathcal{Q}_{p}$, i.e.

$\mathcal{C}(p)=\\{\omega\in T_{p}X:\mathcal{Q}_{p}(\omega)\geq 0\\}.$

The coefficient $\beta$ of expansion at $\omega\in T_{p}X$ is computed as

$\beta(\omega,d_{p}T)=\sqrt{\frac{\mathcal{Q}_{p}(d_{p}T\omega)}{\mathcal{Q}_{p}(\omega)}}$

and the _least coefficient $\sigma$ of expansion_ is defined as

$\sigma(d_{p}T)=\inf_{\omega\in int\mathcal{C}(p)}\beta(\omega,d_{p}T)$

Now we list here the six conditions of [20] in two dimensional case.

  1. (1)

The phase space $X$ is a finite disjoint union of compact subsets of a linear
symplectic space $\mathbb{R}^{2}$ with dense and connected interior and
_regular_ boundaries, i.e. they are finite unions of curves which intersect
each other at at most finitely many points.

  2. (2)

For every $n\geq 1$ the singularity sets $\mathcal{S}_{n}^{+}$ and
$\mathcal{S}_{n}^{-}$ of $T^{n}$ and $T^{-n}$ respectively are regular.

  3. (3)

Almost every point $p\in X$ possesses strictly monotone cones $\mathcal{C}(p)$
and its complementary $\mathcal{C}^{\prime}(p)$.

  4. (4)

The singularity sets $\mathcal{S}^{+}$ and $\mathcal{S}^{-}$ are properly
aligned, i.e. the tangent line of $\mathcal{S}^{-}$ at any
$p\in\mathcal{S}^{-}$ is contained strictly in the cone $\mathcal{C}(p)$ and
the tangent line of $\mathcal{S}^{+}$ at any $p\in\mathcal{S}^{+}$ is
contained strictly in the complementary cone $\mathcal{C}^{\prime}(p)$. In
fact, it is sufficient to assume that there exists $N$ such that
$T^{N}\mathcal{S}^{-}$ and $T^{-N}\mathcal{S}^{+}$ are properly aligned.

  5. (5)

_Noncontraction_ : There is a constant $a\in(0,1]$ such that for every $n\geq
1$ and for every $p\in X\backslash\mathcal{S}_{n}^{+}$

$||d_{p}T^{n}v||\geq a||v||$

for every vector $v\in\mathcal{C}(p)$.

  6. (6)

_Sinai-Chernov Ansatz_ : For almost every $p\in\mathcal{S}^{-}$ with respect
to the measure $\mu_{\mathcal{S}}$, its least coefficient of expansion
satisfies

$\lim_{n\to\infty}\sigma(d_{p}T^{n})=\infty.$

We note from [20] that $\sigma$ is supermultiplicative, i.e.
$\sigma(L_{1}L_{2})\geq\sigma(L_{1})\sigma(L_{2})$, and that if the
coordinates are such that the cone $\mathcal{C}(p)$ is the positive cone and
$d_{p}T$ takes the form

$d_{p}T=\begin{pmatrix}A&B\\\ C&D\end{pmatrix}$

then $\sigma$ can be computed as $\sigma(L)=\sqrt{1+t}+\sqrt{t}$ where $t=BC$.

Liverani and Wojtkowski have proved local ergodicity for symplectic maps
satisfying the above six conditions:

######  Theorem 9 ([20]).

Suppose $(T,X)$ satisfies the above conditions. For any $n\geq 1$ and for any
$p\in X\backslash\mathcal{S}_{n}^{+}$ such that $\sigma(d_{p}T^{n})>3$ there
is a neighborhood of $p$ which is contained in one ergodic component of $T$.

Now we prove Theorem 1:

###### Proof of Theorem 1.

Suppose that $\ddot{f}>0$.

First we prove local ergodicity by verifying the above six conditions for
$\tilde{F}_{\infty}$.

The singularity lines $\mathcal{S}^{\pm}$ are finite unions of short
lines/curves and they cut our phase space $X$, which is a torus, into finitely
many pieces.

The strict monotonicity follows easily from the fact that
$d\tilde{F}_{\infty}$ is positive when $\ddot{f}>0$, hence
$d_{p}\tilde{F}_{\infty}$ preserves strictly the positive cone
$\mathcal{C}^{+}(p)=\\{\delta\tilde{t}\delta\tilde{v}\geq 0\\}$ and
$d_{p}\tilde{F}_{\infty}^{-1}$ preserves strictly the complementary negative
cone $\mathcal{C}^{-}(p)=\\{\delta\tilde{t}\delta\tilde{v}\leq 0\\}$.

It is straightforward from the previous discussion that $\mathcal{S}^{\pm}$
are properly aligned since the slope of tangent line to
$\\{\tilde{t}_{1}=0\\}$ at $(\tilde{t}_{0},\tilde{v}_{0})$ is
$\displaystyle-\frac{g}{2}<0$, and the slope of the tangent line to
$\\{\tilde{t}_{-1}=0\\}$ is
$\displaystyle\frac{g}{2}\left(1+\frac{4k_{0}}{g}\right)>0$.

Next we verify the noncontraction property.
For any nonsingular point $p=(\tilde{t},\tilde{v})$, any vector
$\bm{v}=(\delta\tilde{t},\delta\tilde{v})\in\mathcal{C}^{+}(p)$ with
$\delta\tilde{t}\delta\tilde{v}\geq 0$,

(2)
$\begin{split}||d_{p}\tilde{F}_{\infty}\bm{v}||^{2}&=(1+4k_{1}^{2})\delta\tilde{t}^{2}+\left(\frac{4}{g^{2}}+\left(\frac{4k_{1}}{g}+1\right)^{2}\right)\delta\tilde{v}^{2}+\left(\frac{4}{g}+4k_{1}\left(\frac{4k_{1}}{g}+1\right)\right)\delta\tilde{t}\delta\tilde{v}\\\
&\geq\delta\tilde{t}^{2}+\delta\tilde{v}^{2}=||\bm{v}||^{2}\end{split}$

thus the noncontraction property follows for $a=1$.

Now we verify the Sinai-Chernov Ansatz.
We denote as $\mathcal{S}_{0}=\\{\tilde{t}_{0}=0\\}$. For any point
$p\in\mathcal{S}^{-}\backslash\cup_{n\geq 0}\mathcal{S}_{n}$, which excludes a
$\tilde{\mu}_{\mathcal{S}^{-}}$-null set since $\mathcal{S}^{-}$ intersect
each $\mathcal{S}_{n}$ at at most finitely many points,

$\sigma(d_{p}\tilde{F}_{\infty})=\sqrt{1+\frac{4k_{1}}{g}}+\sqrt{\frac{4k_{1}}{g}}\geq\sqrt{1+\frac{4k_{\min}}{g}}+\sqrt{\frac{4k_{min}}{g}}>1,$

where $\displaystyle k_{\min}=\min_{t\in(0,1)}\ddot{f}(t)>0$ by our
assumption, then the supermultiplicativity of $\sigma$ implies that
$\displaystyle\lim_{n\to\infty}\sigma(d_{p}\tilde{F}_{\infty}^{n})=\infty$.

Finally, it remains to check that the singularity sets $\mathcal{S}_{n}^{-}$
and $\mathcal{S}_{n}^{+}$ of $\tilde{F}_{\infty}^{n}$ and
$\tilde{F}_{\infty}^{-n}$ respectively are regular. We claim that for every
$n>0$, $\mathcal{S}_{n}^{-}$ ($\mathcal{S}_{n}^{+}$) is a finite union of
increasing (decreasing) curves, i.e. curves with bounded positive (negative)
slope. This can be proved by an inductive argument. Firstly the claim holds
for $n=1$ as already shown before. Now suppose $\mathcal{S}_{n}^{-}$ is a
finite union of increasing curves. Since
$\mathcal{S}_{n+1}^{-}=\mathcal{S}_{n}^{-}\cup\tilde{F}_{\infty}\mathcal{S}_{n}^{-}$
and $d\tilde{F}_{\infty}$ is a positive matrix and the second derivative
$\ddot{f}$ is bounded, $\mathcal{S}_{n+1}^{-}$ is a finite union of increasing
curves. The claim for $\mathcal{S}_{n}^{+}$ can be proved similarly.

Observe that $k(t)=\ddot{f}(t)>0$ is uniformly bounded, hence there exists
$N>0$ such that $\sigma(d_{p}\tilde{F}_{\infty}^{N})>3$ for any
$p\notin\mathcal{S}^{+}_{N}$. Therefore we have obtained local ergodicity for
$\tilde{F}_{\infty}$ by Theorem 9.

Now we argue for global ergodicity by contradiction.
Suppose that there exists some nontrivial ergodic component $M$ of
$\tilde{F}_{\infty}$, then its boundary $\partial M$ lies on
$\mathcal{S}^{+}_{N}$. But

$\mathcal{S}^{+}_{N}=\bigcup_{n=0}^{N-1}\tilde{F}_{\infty}^{-n}\mathcal{S}^{+}$

hence there exists a smallest integer $N_{0}\geq 1$ such that $\partial
M\in\mathcal{S}^{+}_{N_{0}}$.
Observe that $\tilde{F}_{\infty}(\partial M)=\partial M$ by the invariance of
$M$. However,
$\tilde{F}_{\infty}(\mathcal{S}^{+}_{N_{0}})=\mathcal{S}^{+}_{N_{0}-1}\cup\mathcal{S}_{0}$,
which contradicts the minimality of $N_{0}$. Note that although
$\tilde{F}_{\infty}$ is multivalued at $\mathcal{S}^{+}$, we have
$\tilde{F}_{\infty}\mathcal{S}^{+}=\mathcal{S}_{0}$ anyway.

Therefore we conclude that there cannot be any nontrivial ergodic component
and hence $\tilde{F}_{\infty}$ is ergodic. ∎

The proof of Theorem 2 follows a similar strategy. The main difficulty arises
from finding invariant cones as the derivative matrix is no longer positive.
However we still manage to construct invariant cones out of the “eigenvectors”
of the derivative matrix.

###### Proof of Theorem 2.

Suppose $\ddot{f}<-g$.
First of all, we recall that the derivative of $\tilde{F}_{\infty}$ at
$(\tilde{t}_{0},\tilde{v}_{0})$ is

$d_{(\tilde{t}_{0},\tilde{v}_{0})}\tilde{F}_{\infty}=\begin{pmatrix}1&\frac{2}{g}\\\
2k_{1}&\frac{4k_{1}}{g}+1\end{pmatrix}.$

Now we consider the following two cones

$\mathcal{C}^{u}(\tilde{t}_{0},\tilde{v}_{0})=\left\\{\frac{\delta v}{\delta
t}\leq
k_{0}\right\\},\quad\mathcal{C}^{s}(\tilde{t}_{0},\tilde{v}_{0})=\left\\{\frac{\delta
v}{\delta t}\geq k_{0}\right\\}$

We verify that they are invariant under
$d_{(\tilde{t}_{0},\tilde{v}_{0})}\tilde{F}_{\infty}$ and
$d_{(\tilde{t}_{0},\tilde{v}_{0})}\tilde{F}_{\infty}^{-1}$ respectively.
For any $(\delta t,\delta v)\in\mathcal{C}^{u}(\tilde{t}_{0},\tilde{v}_{0})$,

$\begin{pmatrix}\bar{\delta t}\\\ \bar{\delta
v}\end{pmatrix}=\begin{pmatrix}1&\frac{2}{g}\\\
2k_{1}&\frac{4k_{1}}{g}+1\end{pmatrix}\begin{pmatrix}\delta t\\\ \delta
v\end{pmatrix}=\begin{pmatrix}\delta t+\frac{2}{g}\delta v\\\ 2k_{1}(\delta
t+\frac{2}{g}\delta v)+\delta v\end{pmatrix}$

thus

(3) $\begin{split}\frac{\bar{\delta v}}{\bar{\delta
t}}&=2k_{1}+\frac{\frac{\delta v}{\delta t}}{1+\frac{2}{g}\frac{\delta
v}{\delta t}}\\\ &\leq 2k_{1}+\frac{k_{0}}{1+\frac{2k_{0}}{g}}\\\
&<2k_{1}+g<k_{1}\end{split}$

where the first inequality follows from $\frac{\delta v}{\delta t}\leq k_{0}$
and the last two inequalities from $\ddot{f}<-g$. Thus $(\bar{\delta
t},\bar{\delta v})\in\mathcal{C}^{u}(\tilde{t}_{1},\tilde{v}_{1})$.
For any $(\delta t,\delta v)\in\mathcal{C}^{s}(\tilde{t}_{0},\tilde{v}_{0})$

$\begin{pmatrix}\tilde{\delta t}\\\ \tilde{\delta
v}\end{pmatrix}=\begin{pmatrix}\frac{4k_{0}}{g}+1&-\frac{2}{g}\\\
-2k_{0}&1\end{pmatrix}\begin{pmatrix}\delta t\\\ \delta
v\end{pmatrix}=\begin{pmatrix}\delta t-\frac{2}{g}(-2k_{0}\delta t+\delta
v)\\\ -2k_{0}\delta t+\delta v\end{pmatrix},$

thus

(4) $\begin{split}\frac{\tilde{\delta v}}{\tilde{\delta
t}}&=-\frac{g}{2}+\frac{g}{2}\frac{1}{1+\frac{2}{g}(2k_{0}-\frac{\delta
v}{\delta t})}\\\ &>-\frac{g}{2}+\frac{g}{2}\frac{1}{1+\frac{2k_{0}}{g}}\\\
&>-g>k_{-1}\end{split}$

where the first inequality follows from $\frac{\delta v}{\delta t}\geq k_{0}$
and the last two inequalities from $\ddot{f}<-g$. Thus $(\tilde{\delta
t},\tilde{\delta v})\in\mathcal{C}^{u}(\tilde{t}_{-1},\tilde{v}_{-1})$.

If we can verify that $\mathcal{S}^{\pm}$ are properly aligned, then the
regularity of the singularity curves $\mathcal{S}_{n}^{\pm}$ follows
automatically from the strict invariance of the cones $\mathcal{C}^{u/s}$
since $\mathcal{S}_{n}^{\pm}$ consist of finitely many transverse short
curves.
We claim that $\mathcal{S}^{\pm}$ are properly aligned. Indeed, the slope of
tangent line to $\\{\tilde{t}_{1}=0\\}$ at $(\tilde{t}_{0},\tilde{v}_{0})$ is
$\displaystyle-\frac{g}{2}>k_{0}$, which is properly contained in
$\mathcal{C}^{s}$. Also, the slope of the tangent line to
$\\{\tilde{t}_{-1}=0\\}$ is $\displaystyle 2k_{0}+\frac{g}{2}<k_{0}$, which is
properly contained in $\mathcal{C}^{u}$.

The noncontraction property still holds with $a=1$.

The unstable cone $\mathcal{C}^{u}$ is not canonical, i.e. it is not the
positive cone, hence we need to switch to the new basis $((0,1),(1,k_{0}))$
and $d_{p}\tilde{F}_{\infty}$ takes the form

$\begin{pmatrix}\frac{2k_{1}}{g}+1&k_{0}+k_{1}+\frac{2k_{0}k_{1}}{g}\\\
2/g&1+\frac{2k_{0}}{g}\end{pmatrix}$

Then the Sinai-Chernov Ansatz holds since

$\displaystyle\sigma(d_{p}\tilde{F}_{\infty})$
$\displaystyle=\sqrt{1+\frac{2}{g}\left(\frac{2k_{0}k_{1}}{g}+k_{0}+k_{1}\right)}+\sqrt{\frac{2}{g}\left(\frac{2k_{0}k_{1}}{g}+k_{0}+k_{1}\right)}$
$\displaystyle\geq\sqrt{1+\frac{2}{g}\left(\frac{2k_{\min}^{2}}{g}+2k_{\min}\right)}+\sqrt{\frac{2}{g}\left(\frac{2k_{\min}^{2}}{g}+2k_{\min}\right)}>1.$

By Theorem 9 the local ergodicity of $\tilde{F}_{\infty}$ for the case when
$\ddot{f}<-g$. Finally, the global ergodicity can be obtained by a similar
argument as in the proof of Theorem 1. ∎

##  5\. Recurrence of the Collision Map

In this section, we prove Theorem 5 and Theorem 6 as they are direct
consequences of the ergodicity of the limit map $\tilde{F}_{\infty}$ on the
torus.
The proof of Theorem 5 uses a result from [8], which shows that for an
asymptotically periodic map with an ergodic limiting map, if the energy change
in the limiting map has zero average, then the escaping orbits of the original
dynamics have zero measure.

We state this result for our case specifically. First we decompose the
velocity $v$ into integer part and fractional part, i.e. there exists some
$m\in\mathbb{Z}$ such that

$v=\tilde{v}+mg\text{ where }\tilde{v}\in[0,g).$

Then we decompose the limit map $F_{\infty}$ on the cylinder into its
projection $\tilde{F}_{\infty}$ on the torus and a map $\gamma$ on integers
$\mathbb{Z}$, i.e.

$(\tilde{t}_{1},\tilde{v}_{1},m_{1})=F_{\infty}(\tilde{t}_{0},\tilde{v}_{0},m_{0})=(\tilde{F}_{\infty}(\tilde{t}_{0},\tilde{v}_{0}),m_{0}+\gamma(\tilde{t}_{0},\tilde{v}_{0})).$

######  Lemma 5.1 ([8]).

Suppose that $\tilde{F}_{\infty}$ is ergodic with respect to the measure
$\tilde{\mu}=d\tilde{t}d\tilde{v}$ on the torus. If the energy change of
$F_{\infty}$ has zero average, i.e.
$\int_{\mathbb{T}}\gamma(\tilde{t}_{0},\tilde{v}_{0})d\tilde{\mu}=0$, then the
escaping set of $F$ has zero measure.

Now we prove Theorem 5.

###### Proof of Theorem 5.

If $f(t)$ is admissible, then, by Theorems 1 and 2, $\tilde{F}_{\infty}$ is
ergodic. Thus by Lemma 5.1, it suffices to check that the energy change
$\gamma$ of $F_{\infty}$ has zero average.
With an abuse of notation, let us denote
$(t_{1},v_{1})=F_{\infty}(t_{0},v_{0})$ and
$(\tilde{t}_{1},\tilde{v}_{1})=\tilde{F}_{\infty}(\tilde{t}_{0},\tilde{v}_{0})$
respectively. Observe that

$\int\tilde{v}_{0}d\tilde{\mu}=\int\tilde{v}_{1}d\tilde{\mu}$

since $\tilde{F}_{\infty}$ preserves the measure $\tilde{\mu}$. Thus

$\int\gamma d\tilde{\mu}=\frac{1}{g}\int(v_{1}-v_{0})d\tilde{\mu}.$

But $v_{1}-v_{0}=2\dot{f}(\tilde{t}_{1})$, hence

$\int(v_{1}-v_{0})d\tilde{\mu}=\int 2\dot{f}(\tilde{t}_{1})d\tilde{\mu}=\int
2\dot{f}(\tilde{t}_{0})d\tilde{\mu}=0$

where the last two equalities follow from the fact that $\tilde{F}_{\infty}$
preserves $\tilde{\mu}$ and that $f$ is 1-periodic. Therefore by Lemma 5.1 the
escaping orbits of $F$ have zero measure. ∎

The set $E$ of escaping orbits is in fact the transient part of $F$, hence
Theorem 5 implies that $F$ is recurrent, in the spirit of [10]:

###### Proof of Corollary 6.

We claim that the set $E$ of escaping orbits is the transient component of the
system. Hence Theorem 5 implies Corollary 6.

Indeed, the complement of $E$ is $\cup E_{N}$ where

$E_{N}=\\{(t_{0},v_{0}):\liminf v_{n}\leq N\\}.$

For any $N\in\mathbb{N}$, $E_{N}$ is invariant and all points in $E_{N}$ will
visit the set $V_{N}=\\{v\leq N+1\\}$.
Suppose $A$ is a subset of $E_{N}$ with finite measure. For any $x\in A$, we
denote the first hitting time in $V_{N}$ as $r(x)=\min\\{k\geq 0:F^{k}x\in
V_{N}\\}$. Now for any $K\in\mathbb{N}$, we consider

$A_{K}:=\bigcup_{x\in A:r(x)\leq K}F^{r(x)}x$

To show recurrence in $A$, it suffices to prove that almost every point in
$A_{K}$ visits itself infinitely often since if for $x\in A_{K}$ $F^{n}x\in
A_{K}$ for some $n>K$, then there exists some $x^{\prime}\in A$ such that
$F^{n-r(x^{\prime})}x=x^{\prime}\in A$.
However $A_{K}\subseteq E_{N}\cap V_{N}$ by definition of $A_{K}$ and the
invariance of $E_{N}$. All points in $E_{N}$ visit $V_{N}$, thus the first
return map $P$ on $A_{K}$ is well-defined. Now our goal is achieved by
applying Poincaré recurrence theorem to $(A_{K},P)$. ∎

##  6\. Statistical Properties of the Limit Map

In this section we prove Theorem 3, Theorem 4 and Theorem 8. Throughout this
section we assume the wall motion is admissible.

###  6.1. Background.

The proof of Theorem 3 uses a result of Chernov and Zhang in [6] and the proof
of Theorem 4 uses a result of Chernov in [4]. We first describe the class of
hyperbolic symplectic maps considered in [4, 6] and then show that our map
$\tilde{F}_{\infty}$ belongs to this class.

Let $T:M\to M$ be a $C^{2}$ diffeomorphism of a two dimensional Riemannian
manifold $M$ with singularities $\mathcal{S}$. Suppose $T$ satisfies the
following conditions:

  1. (1)

_Uniform hyperbolicity of $T$_. There exist two continuous families of
unstable cones $\mathcal{C}_{x}^{u}$ and stable cones $\mathcal{C}_{x}^{s}$ in
the tangent spaces $\mathcal{T}_{x}M$ for all $x\in M$, and $\exists$ a
constant $\Lambda>1$ such that

    1. (a)

$DT(\mathcal{C}_{x}^{u})\subset\mathcal{C}_{Tx}^{u}$, and
$DT(\mathcal{C}_{x}^{s})\supset\mathcal{C}_{Tx}^{s}$ whenever $DT$ exists;

    2. (b)

$||D_{x}T\rm{v}||\geq\Lambda||\rm{v}||$ $\forall\rm{v}\in\mathcal{C}_{x}^{u}$,
and $||D_{x}T^{-1}\rm{v}||\geq\Lambda||\rm{v}||$
$\forall\rm{v}\in\mathcal{C}_{x}^{s}$;

    3. (c)

The angle between $\mathcal{C}_{x}^{u}$ and $\mathcal{C}_{x}^{s}$ is uniformly
bounded away from zero.

  2. (2)

_Singularities $\mathcal{S}^{\pm}$ of $T$ and $T^{-1}$_. The singularities
$\mathcal{S}^{\pm}$ have the following properties:

    1. (a)

$T:M\backslash\mathcal{S}^{+}\to M\backslash\mathcal{S}^{-}$ is a $C^{2}$
diffeomorphism;

    2. (b)

$\mathcal{S}_{0}\cup\mathcal{S}^{+}$ is a finite or countable union of smooth
compact curves in $M$;

    3. (c)

Curves in $\mathcal{S}_{0}$ are transverse to the stable and unstable cones.
Every smooth curve in $\mathcal{S}^{+}$ ($\mathcal{S}^{-}$) is a stable
(unstable) curve. Every curve in $\mathcal{S}^{+}$ terminates either inside
another curve of $\mathcal{S}^{+}$ or on $\mathcal{S}_{0}$;

    4. (d)

$\exists\ \beta\in(0,1)$ and $c>0$ such that for any $x\in
M\backslash\mathcal{S}^{+}$, $||D_{x}T||\leq cd(x,\mathcal{S}^{+})^{-\beta}$.

  3. (3)

_Regularity of smooth unstable curves._ We assume there exists a $T$-invariant
class of unstable curves $W$ such that

    1. (a)

_Bounded curvature._ The curvature of $W$ is uniformly bounded from above;

    2. (b)

_Distortion control._ $\exists\ \gamma\in(0,1)$ and $C>1$ such that for any
regular unstable curve $W$ and any $x,y\in W$

$\big{|}\log\mathcal{J}_{W}(x)-\log\mathcal{J}_{W}(y)\big{|}\leq
Cd(x,y)^{\gamma},$

where $\mathcal{J}_{W}(x)=|D_{x}T|_{W}|$ denotes the Jacobian of $T$ at $x\in
W$;

    3. (c)

_Absolute continuity of the holonomy map._ Let $W_{1},W_{2}$ be two regular
unstable curves that are close to each other. We denote

$W_{i}^{\prime}=\\{x\in W_{i}:W^{s}(x)\cap W_{3-i}\neq\emptyset\\},\ \ i=1,2.$

The holonomy map $h:W_{1}^{\prime}\to W_{2}^{\prime}$ is defined by sliding
along the stable manifold. We assume that
$h_{\star}\mu_{W_{1}^{\prime}}\ll\mu_{W_{2}^{\prime}}$ and that for some
constant $C$ and $\vartheta<1$

$\big{|}\log\mathcal{J}h(x)-\log\mathcal{J}h(y)\big{|}\leq
C\vartheta^{s_{+}(x,y)},\quad x,y\in W_{1}^{\prime}$

where $\mathcal{J}h$ is the Jacobian of $h$;

  4. (4)

_SRB measure_. $\tilde{\mu}$ is an SRB measure, i.e. the induced measure
$\tilde{\mu}_{W^{u}}$ on any unstable manifold $W^{u}$ is absolutely
continuous with respect to $Leb_{W^{u}}$. We also assume that $\tilde{\mu}$
and is mixing.

  5. (5)

_One-step expansion._ Let $\xi^{n}$ denote the partition of $M$ into connected
components of $M\backslash\mathcal{S}_{n}^{+}$. Denote as $V_{\alpha}$ the
connected component of $TW$ with index $\alpha\in M/\xi^{1}$ and
$W_{\alpha}=T^{-1}V_{\alpha}$. $\exists\ q\in(0,1]$ such that

$\liminf_{\delta\to 0}\sup_{W:|W|<\delta}\sum_{\alpha\in
M/\xi^{1}}\bigg{(}\frac{|W|}{|V_{\alpha}|}\bigg{)}^{q}\frac{|W_{\alpha}|}{|W|}<1,$

where the supremum is taken over all unstable curves $W$.

######  Theorem 10 ([4, 6]).

Under the assumptions 1-5 above, the system $(T,M)$ above enjoys exponential
decay of correlations and central limit theorem for dynamically Hölder
continuous observables.

The verifications of Assumptions 1-5 are rather long. Moreover, their validity
is of independent importance themselves. So we first state intermediary
lemmata in Section 6.2, then we prove, based on these lemmata, Theorem 3,
Theorem 4 and Theorem 8 in Section 6.3, and finally we prove all the lemmata
in Section 6.4.

###  6.2. Intermediary Lemmas

In this section we list the intermediate lemmata. Their proofs are presented
in Section 6.4.

Suppose $W$ is an unstable curve, i.e. the tangent line of $W$ lies in the
unstable cone $\mathcal{C}^{u}$, with bounded curvature. We assume without
loss of generality that $W\cap\mathcal{S}^{+}=\emptyset$. Let
$\mathcal{J}_{W}(x)=|D_{x}T|_{W}|$ denote the Jacobian of $\tilde{F}_{\infty}$
at $x\in W$. We have the following enhanced distortion control:

######  Lemma 6.1 (Distortion Control).

Suppose $W$ is an unstable curve with bounded curvature. Then for any $x,y\in
W$, there exists a constant $C$ which depends only on $\tilde{F}_{\infty}$
such that

$\big{|}\log\mathcal{J}_{W}(x)-\log\mathcal{J}_{W}(y)\big{|}\leq Cd(x,y).$

Furthermore, if $W\cap\mathcal{S}_{N}^{-}=\emptyset$, then for any $1\leq
n\leq N$ there exists a constant $C^{\prime}$ which depends only on
$\tilde{F}_{\infty}$ such that

$\big{|}\log\mathcal{J}_{W}\tilde{F}_{\infty}^{-n}(x)-\log\mathcal{J}_{W}\tilde{F}_{\infty}^{-n}(y)\big{|}\leq
C^{\prime}|W|.$

In order to establish the $N_{0}$-step expansion, we need the following
estimate on the speed of fragmentation of unstable curves:

######  Lemma 6.2 (Complexity Bound).

Suppose $z$ is a multiple point of $\mathcal{S}_{n}^{+}$. Pick a small
neighborhood of $z$ and denote as $k_{n}(z)$ the number of sectors in the
small neighborhood cut out by $\mathcal{S}_{n}^{+}$. Then $k_{n}(z)\leq 6n$.

The linear complexity bound guarantees that a sufficiently short unstable
curve $W$ can break into at most $6n$ connected components under
$\tilde{F}_{\infty}^{n}$. Thus there exists $\delta_{0}$ so small that any
unstable curve shorter than $\delta_{0}$ satisfies the following expansion
estimate:

######  Lemma 6.3 ($N_{0}$-Step Expansion).

Suppose that $W$ is an unstable curve with length $|W|\leq\delta_{0}$ and that
$\\{W_{i}^{n}\\}_{i}$ are the connected components of the image
$\tilde{F}_{\infty}^{n}W$. Denote as $\Lambda_{i,n}$ the minimum rate of
expansion on each preimage $\tilde{F}_{\infty}^{-n}W_{i}^{n}$. Then

$\sum_{i}\frac{1}{\Lambda_{i,N_{0}}}<1,$

where $N_{0}$ is the smallest integer such that
$\displaystyle\frac{6N_{0}}{\Lambda^{N_{0}}}<1$ and $\Lambda$ is the expansion
rate of $\tilde{F}_{\infty}$.

Next, we suppose that $W$ and $\bar{W}$ are two unstable curves with bounded
curvatures. We define the following holonomy map $h$ on

$W^{\prime}=\\{x\in W:W^{s}(x)\cap\bar{W}\neq\emptyset\\}$

by sliding along the stable manifold from $x\in W$ to $\bar{x}\in\bar{W}$.
Then $h:W^{\prime}\to\bar{W}^{\prime}$ is absolutely continuous with well-
behaving density:

######  Lemma 6.4 (Absolute Continuity).

Suppose $W$ and $\bar{W}$ are two unstable curves with bounded curvatures.
Then $h_{\star}\mu_{W}\ll\mu_{\bar{W}}$ and that for some constant $C$ and
$\Theta<1$

$\big{|}\log\mathcal{J}h(x)-\log\mathcal{J}h(y)\big{|}\leq
C\Theta^{s_{+}(x,y)},\quad x,y\in W^{\prime}$

where $\mathcal{J}h$ is the Jacobian of $h$.

Finally we provide an estimate on the number of small unstable curves, which
follows from Lemma 7 in [6].
Suppose $W$ is an unstable curve with length $|W|<\delta_{0}$. For any $x\in
W$, we denote as $r_{n}(x)$ the distance from $x$ to the nearest boundary of
the connected component of $\tilde{F}_{\infty}^{n}W$ containing
$\tilde{F}_{\infty}^{n}x$.

######  Lemma 6.5 (Growth Lemma).

Suppose $W$ is an unstable curve with length $|W|<\delta_{0}$. Then for any
$\epsilon>0$,

$\text{mes}_{W}\\{r_{nN_{0}}(x)<\epsilon\\}\leq(\vartheta_{1}\Lambda^{N_{0}})^{n}\text{mes}_{W}\left\\{r_{0}(x)<\frac{\epsilon}{\Lambda^{nN_{0}}}\right\\}+C\epsilon|W|$

where
$\vartheta_{1}=e^{C^{\prime}\delta_{0}}\sum_{i}\frac{1}{\Lambda_{i,N_{0}}}<1$,
$C^{\prime}$ is the constant from Lemma 6.1, $N_{0}$ is the constant from
Lemma 6.3 and $\Lambda$ is the expansion rate of $\tilde{F}_{\infty}$.

######  Remark 6.1.

We note that $\vartheta_{1}$ can be made less than 1 by choosing $\delta_{0}$
sufficiently small.

###  6.3. Exponential Decay of Correlations, CLT and Global Global Mixing

In this section we present the proof of Theorem 3, Theorem 4 and Theorem 8,
based on the lemmas from Section 6.2.

We start with the proof for exponentially decay of correlations and CLT.

###### Proof of Theorem 3 and Theorem 4.

Firstly we establish the exponential decay and CLT for
$\tilde{F}_{\infty}^{N_{0}}$ by checking the conditions in Theorem 10 for
$\tilde{F}_{\infty}^{N_{0}}$ where $N_{0}$ is the number from Lemma 6.3. Note
that we gain from Theorem 10 the exponential decay and CLT for
$\tilde{F}_{\infty}^{N_{0}}$ rather than for $\tilde{F}_{\infty}$ because we
can only obtain $N_{0}$-step expansion on $\tilde{F}_{\infty}$.
The proof for the case $\ddot{f}>0$ is very similar to that for $\ddot{f}<-g$,
and thence we omit the latter. Although the positive/negative cones in the
proof of Theorem 1 are strictly invariant, we cannot use them here since we
require a positive angle between the unstable and stable cones. Instead, we
consider their images, i.e.

$\mathcal{C}^{u}(\tilde{t}_{0},\tilde{v}_{0})=\left\\{2k_{0}\leq\frac{\delta
v}{\delta t}\leq 2k_{0}+\frac{g}{2}\right\\}$
$\mathcal{C}^{s}(\tilde{t}_{0},\tilde{v}_{0})=\left\\{-\frac{g}{2}\leq\frac{\delta
v}{\delta t}\leq-\frac{2k_{0}}{\frac{4k_{0}}{g}+1}\right\\}.$

It is easy to see that the angles between $\mathcal{C}^{u}$ and
$\mathcal{C}^{s}$ are uniformly bounded away from zero since $k_{0}>0$ is
bounded. and that these cones are strictly invariant.

We now compute the expansion rate $\Lambda$. With the same notations as above,
for $(\delta t,\delta v)\in\mathcal{C}^{u}(\tilde{t}_{0},\tilde{v}_{0})$ it
follows from (2) that

$\bar{\delta t}^{2}+\bar{\delta v}^{2}\geq\Lambda_{1}^{2}(\delta t^{2}+\delta
v^{2})$

where
$\displaystyle\Lambda_{1}^{2}=\min\left\\{1+4k_{min},\frac{4}{g^{2}}+\left(1+\frac{4k_{min}}{g}\right)^{2}\right\\}>1$.
Similarly for $(\delta t,\delta
v)\in\mathcal{C}^{s}(\tilde{t}_{0},\tilde{v}_{0})$

$\displaystyle\tilde{\delta t}^{2}+\tilde{\delta v}^{2}$
$\displaystyle=\left(4k_{1}^{2}+\left(1+\frac{4k_{1}}{g}\right)^{2}\right)\delta
t^{2}+\left(\frac{4}{g^{2}}+1\right)\delta
v^{2}-2\left(2k_{1}+\frac{2}{g}\left(1+\frac{4k_{1}}{g}\right)\right)\delta
t\delta v$ $\displaystyle\geq\Lambda_{2}^{2}(\delta t^{2}+\delta v^{2})$

where
$\displaystyle\Lambda_{2}^{2}=\min\left\\{4k_{min}^{2}+\left(1+\frac{4k_{min}}{g}\right)^{2},\frac{4}{g^{2}}+1\right\\}>1$.
Take $\Lambda=\min\\{\Lambda_{1},\Lambda_{2}\\}$, and this gives our expansion
rate.

Next we examine the singularity curves $\mathcal{S}^{\pm}$.
Observe that $\tilde{F}_{\infty}$ is a $C^{2}$-diffeomorphism away from
singularities if $f$ is piecewise $C^{3}$. And
$\mathcal{S}_{0}\cup\mathcal{S}^{+}$ is a finite union of smooth compact
curves on the torus $\mathbb{T}$. $\mathcal{S}_{0}$ is transverse to
$\mathcal{C}^{u}/\mathcal{C}^{s}$. Moreover the singularity curves are regular
and properly aligned as shown in the proof of Theorem 1.
Assumption 2(d) is trivially satisfied since the norm of the derivative
$d\tilde{F}_{\infty}$ is bounded and our phase space is compact.

As for Assumption 3, we have already obtained distortion control in Lemma 6.1
and absolute continuity of holonomy map in Lemma 6.4. We note that by (6) the
curvature of an unstable curve remains bounded after iterations.

Next, the invariant measure $\tilde{\mu}=d\tilde{t}d\tilde{v}$ is apparently
an SRB measure.
Note that $\tilde{F}_{\infty}^{n}$ is ergodic with respect to $\tilde{\mu}$
for any $n>0$, since $\tilde{F}_{\infty}^{n}$ also satisfies the conditions of
Theorem 9. Now the results of [23] imply that $\tilde{\mu}$ is mixing (even
Bernoulli).

Finally, since we already establish $N_{0}$-step expansion from Lemma 6.3, we
conclude from Theorem 10 that $\tilde{F}_{\infty}^{N_{0}}$ enjoys exponential
decay of correlations and CLT for dynamically Hölder continuous observables.

The CLT for $\tilde{F}_{\infty}$ follows easily from that of
$\tilde{F}_{\infty}^{N_{0}}$. Now we show that exponential mixing for
$\tilde{F}_{\infty}^{N_{0}}$ implies exponential mixing for
$\tilde{F}_{\infty}$.
Suppose $\varphi,\phi$ are two dynamically Hölder continuous observables. For
any integer $n\in\mathbb{N}$, $n=pN_{0}+q$ for some integers $p>0$ and $0\leq
q<N_{0}$.
We denote as $\tilde{\varphi}_{q}=\varphi\circ\tilde{F}_{\infty}^{q}$. For any
$x,y$ on a same unstable manifold $W^{u}$,

$\big{|}\tilde{\varphi}_{q}(x)-\tilde{\varphi}_{q}(y)\big{|}=\big{|}\varphi(\tilde{F}_{\infty}^{q}x)-\varphi(\tilde{F}_{\infty}^{q}y)\big{|}\leq
C\vartheta^{-q_{+}}\vartheta^{s_{+}(x,y)}$

where $q_{+}=\min\\{q,s_{+}(x,y)\\}$.
On the other hand, for any $x,y$ on a same stable manifold $W^{s}$,

$\big{|}\tilde{\varphi}_{q}(x)-\tilde{\varphi}_{q}(y)\big{|}=\big{|}\varphi(\tilde{F}_{\infty}^{q}x)-\varphi(\tilde{F}_{\infty}^{q}y)\big{|}\leq
C\vartheta^{q}\vartheta^{s_{-}(x,y)}.$

Therefore $\tilde{\varphi}_{q}$ is also dynamically Hölder.
By applying the previous exponential decay result on
$\tilde{F}_{\infty}^{N_{0}}$ with the observables $\tilde{\varphi}_{q},\phi$,
we know that $\exists$ $C_{\tilde{\varphi}_{q},\phi}$ and $b$ such that

$\displaystyle\bigg{|}\int_{\mathbb{T}}\big{(}\varphi\circ\tilde{F}_{\infty}^{n}\big{)}\phi
d\tilde{\mu}-\int_{\mathbb{T}}\varphi d\tilde{\mu}\int_{\mathbb{T}}\phi
d\tilde{\mu}\bigg{|}$
$\displaystyle=\bigg{|}\int_{\mathbb{T}}\big{(}\tilde{\varphi}_{q}\circ\tilde{F}_{\infty}^{pN_{0}}\big{)}\phi
d\tilde{\mu}-\int_{\mathbb{T}}\tilde{\varphi}_{q}d\tilde{\mu}\int_{\mathbb{T}}\phi
d\tilde{\mu}\bigg{|}$ $\displaystyle\leq
C_{\tilde{\varphi}_{q},\phi}e^{-bp}=C_{\tilde{\varphi}_{q},\phi}e^{bq/N_{0}}(e^{-b/N_{0}})^{n}.$

If we take $\displaystyle
C_{\varphi,\phi}=\max_{q}\\{C_{\tilde{\varphi}_{q},\phi}e^{bq/N_{0}}\\}$ and
replace $b$ with $b/N_{0}$, then we have proved exponential decay of
correlation in the case $\ddot{f}>0$. ∎

Next we prove the global global mixing property for the original collision map
$F$.

###### Proof of Theorem 8.

Under Assumptions 1-5, the limit map $\tilde{F}_{\infty}$ satisfies the
conditions of [3], thus it admits a Young tower with exponential tail. We
recall from Section 3.2 that $\tilde{F}_{\infty}$ well approximates the
original collision map $F$ at infinity. Therefore by Theorem 2.4 and Theorem
2.9 in [14], $F$ is global global mixing. ∎

###  6.4. Proof of Intermediary Lemmas

In this section we prove the lemmas stated in Section 6.2.

We start with the distortion control.

###### Proof of Lemma 6.1.

We parametrize the unstable W as $v=\psi(t)$ for some smooth function $\psi$
such that $\psi^{\prime}(t)\in[2k,2k+g/2]$ and $\psi^{\prime\prime}$ is
bounded.
For $x,y\in W$,
$\displaystyle\left|\log\mathcal{J}_{W}(x)-\log\mathcal{J}_{W}(y)\right|\leq\max_{z\in
W}\left|\frac{d}{dz}\log\mathcal{J}_{W}(z)\right||x-y|$.
For $z\in W$, we take $\rm{v}=(1,\psi^{\prime}(t))\in\mathcal{T}_{z}W$.

$\mathcal{J}_{W}(z)=\frac{\left\|d_{z}\tilde{F}_{\infty}\rm{v}\right\|}{\left\|\rm{v}\right\|}=$
$\frac{1}{\sqrt{1+\psi^{\prime}(t)^{2}}}\left(1+4k_{1}^{2}+\psi^{\prime}(t)^{2}\left(\frac{4}{g^{2}}+\left(1+\frac{4k_{1}}{g}\right)^{2}\right)+2\psi^{\prime}(t)\left(\frac{2}{g}+2k_{1}\left(1+\frac{4k_{1}}{g}\right)\right)\right)^{1/2}$

where $k_{1}=\ddot{f}(\tilde{F}_{\infty}z)$. Hence

$\log\mathcal{J}_{W}(z)=$
$\frac{1}{2}\log\left(1+4k_{1}^{2}+\psi^{\prime}(t)^{2}\left(\frac{4}{g^{2}}+\left(1+\frac{4k_{1}}{g}\right)^{2}\right)+2\psi^{\prime}(t)\left(\frac{2}{g}+2k_{1}\left(1+\frac{4k_{1}}{g}\right)\right)\right)$
(5) $-\frac{1}{2}\log(1+\psi^{\prime}(t)^{2})$

We note that each term inside the logarithms in (5) is greater than one and
has bounded derivatives. Thus
$\left|\frac{d}{dz}\log\mathcal{J}_{W}(z)\right|\leq C$ for some constant $C$
depending only on $\tilde{F}_{\infty}$.
Besides the above distortion bound, we have the following enhanced estimate.
Now we assume further that $W\cap\mathcal{S}^{-}_{n}=\emptyset$.
We denote $x_{n}=\tilde{F}_{\infty}^{-n}x$, $y_{n}=\tilde{F}_{\infty}^{-n}y$
and $W_{n}=\tilde{F}_{\infty}^{-n}W$.

$\displaystyle\left|\log\mathcal{J}_{W}\tilde{F}_{\infty}^{-n}(x)-\log\mathcal{J}_{W}\tilde{F}_{\infty}^{-n}(y)\right|$
$\displaystyle\leq\sum_{m=0}^{n-1}\left|\log\mathcal{J}_{W_{m}}\tilde{F}_{\infty}^{-n}(x_{m})-\log\mathcal{J}_{W_{m}}\tilde{F}_{\infty}^{-n}(y_{m})\right|$
$\displaystyle\leq\sum_{m=0}^{n-1}|W_{m}|\max_{z_{m}\in
W_{m}}\left|\frac{d}{dz_{m}}\log\mathcal{J}_{W_{m}}\tilde{F}_{\infty}^{-1}(z_{m})\right|.$

But

$\displaystyle\frac{d}{dz_{m}}\log\mathcal{J}_{W_{m}}\tilde{F}_{\infty}^{-1}(z_{m})$
$\displaystyle=\frac{dz_{m+1}}{dz_{m}}\frac{d}{dz_{m+1}}\log\frac{1}{\mathcal{J}_{W_{m+1}}\tilde{F}_{\infty}(z_{m+1})}$
$\displaystyle=-\frac{1}{\mathcal{J}_{W_{m+1}}\tilde{F}_{\infty}(z_{m+1})}\frac{d}{dz_{m+1}}\log\mathcal{J}_{W_{m+1}}\tilde{F}_{\infty}(z_{m+1}).$

Observe that $\mathcal{J}_{W_{m+1}}\tilde{F}_{\infty}(z_{m+1})$ is bounded.
Next

$\frac{dv_{m}}{dt_{m}}=\frac{2k_{m}dt_{m-1}+(4k_{m}/g+1)dv_{m-1}}{dt_{m-1}+\frac{2}{g}dv_{m-1}}$

$=\frac{2k_{m}+(4k_{m}/g+1)\frac{dv_{m-1}}{dt_{m-1}}}{1+\frac{2}{g}\frac{dv_{m-1}}{dt_{m-1}}}=2k_{m}+g/2-\frac{g/2}{1+\frac{2}{g}\frac{dv_{m-1}}{dt_{m-1}}}.$

Therefore

$\psi_{m}^{\prime\prime}=2\dddot{f}(t_{m})-\frac{\psi_{m-1}^{\prime\prime}}{\left(1+\frac{2}{g}\psi_{m-1}^{\prime}\right)^{3}}$

which implies that

$|\psi_{m}^{\prime\prime}|\leq
2\dddot{f}_{\max}+\theta|\psi_{m-1}^{\prime\prime}|$

where $\theta:=\frac{1}{(1+4k_{\min}/g)^{3}}<1$. Iterating we obtain

(6)
$|\psi_{m}^{\prime\prime}|\leq\frac{2\dddot{f}_{\max}}{1-\theta}+\theta^{m}|\psi_{0}^{\prime\prime}|.$

Hence
$\displaystyle\left|\frac{d}{dz_{m+1}}\log\mathcal{J}_{W_{m+1}}\tilde{F}_{\infty}(z_{m+1})\right|$
is bounded. Thus

$\left|\log\mathcal{J}_{W}\tilde{F}_{\infty}^{-n}(x)-\log\mathcal{J}_{W}\tilde{F}_{\infty}^{-n}(y)\right|\leq
C^{\prime\prime}\sum_{m=0}^{n-1}|W_{m}|\leq
C^{\prime\prime}\sum_{m=0}^{n-1}\frac{|W|}{\Lambda^{m}}\leq
C^{\prime}|W|.\qed$

Next we prove the complexity bound following an approach of [9].

###### Proof of Lemma 6.2.

Suppose $z$ is a multiple point of $\mathcal{S}_{n}^{+}$. We take a small
neighborhood of $z$ and cut it into four quadrants $Q$’s by vertical and
horizontal lines through $z$. Denote as $k_{n}(z)|_{Q}$ the number of sectors
cut out by $\mathcal{S}_{n}^{+}$ intersecting nontrivially with $Q$.
We are only interested in the active quadrants, i.e. the quadrants in the
northwest and southeast, because the tangent lines to the singularities curves
$\mathcal{S}_{n}^{+}$ have negative slopes and the inactive quadrants (in the
northeast and southwest) remain untouched by them and thus do not contribute
to the complexity growth.

$\mathcal{S}_{n}^{+}$$z$$Q$ Figure 2. A multiple point and its sectors

Denote as $\\{V_{i}\\}$ the sectors cut out by $\mathcal{S}^{+}$. Note that
$\mathcal{S}^{+}=\\{\tilde{t}_{1}=0\\}$, hence there are at most two sectors
cut out by $\mathcal{S}^{+}$ in a quadrant. By further cutting horizontally
and vertically, we might assume each $V_{i}\subseteq Q$.
We denote as $V_{i}^{\prime}=\tilde{F}_{\infty}(V_{i})$,
$z_{i}^{\prime}=\tilde{F}_{\infty}(z_{i})$ (this is defined by continuity),
and $k_{n}(z)|_{Q}=\sum_{i}k_{n-1}(z_{i}^{\prime})|{V_{i}^{\prime}}$.
If $z\notin\mathcal{S}^{+}$, then $i=1$ and
$k_{n}(z)|_{Q}=k_{n-1}(z^{\prime})|{V^{\prime}}$.
If $z\in\mathcal{S}^{+}$, then $i=2$ and we claim that at most one image
$V_{i}^{\prime}$ of the sectors $V_{i}$ remains active, so that in both cases
we have

$\displaystyle k_{n}(z)|_{Q}$
$\displaystyle=\sum_{i}k_{n-1}(z_{i}^{\prime})|{V_{i}^{\prime}}$
$\displaystyle\leq 1+k_{n-1}(z_{i}^{\prime})|{V_{i}^{\prime}}\hbox{
($V_{i}^{\prime}$ is the only active image)}$ $\displaystyle\leq
3+k_{n-1}(z_{i}^{\prime})|{Q_{i}^{\prime}}\hbox{ (by further cutting
$V_{i}^{\prime}$ horizontally and vertically)}$

Thus $k_{n}(z)|_{Q}\leq 3n$ implies $k_{n}(z)\leq 6n$, which is our desired
complexity bound.

Now we prove our claim. Suppose that $z\in\\{\tilde{t}_{1}=0\\}$.

$\tilde{t}_{1}=0$$z$$Q_{NW}$$V_{2}$$V_{1}$$\tilde{F}_{\infty}$$\tilde{t}$$\tilde{v}$$z^{\prime}$$w^{\prime}$$w$
Figure 3. Northwest quadrant for $z\in\\{\tilde{t}_{1}=0\\}$

Recall that $\tilde{t}_{1}=\tilde{t}_{0}+\frac{2\tilde{v}_{0}}{g}$ (mod 1),
$\tilde{v}_{1}=\tilde{v}_{0}+2\dot{f}(\tilde{t}_{1})$ (mod $g$).
Since $z\in\\{\tilde{t}_{1}=0\\}$, the $\tilde{t}$-coordinate of its images
$z^{\prime}$ is zero, i,e. $\tilde{t}(z_{i}^{\prime})=0$ ($i=1,2$).
We pick $w\in V_{1}$ sufficiently close to $z$, then the
$\tilde{t}$-coordinate of its image $w^{\prime}$ is positive since $w$ is at
the right side of the singularity line $\\{\tilde{t}_{1}=0\\}$. Also, since we
assume $\ddot{f}>0$, $\dot{f}$ is increasing and hence the
$\tilde{v}$-coordinate of its image $w^{\prime}$ is larger than that of
$z^{\prime}$. This means that the image
$V_{1}^{\prime}=\tilde{F}_{\infty}(V_{1})$ is inactive.
Similarly, we can show that the lower one to the left of the singularity line
$\mathcal{S}^{+}$ in the southeast quadrant becomes inactive after being
mapped by $\tilde{F}_{\infty}$. ∎

Finally we estimate the Jacobian of the holonomy map.

###### Proof of Lemma 6.4.

It follows from classical results in [1, 27] that the holonomy map is
absolutely continuous and its Jacobian is given by

$\mathcal{J}h(x)=\prod_{j=0}^{\infty}\frac{\mathcal{J}_{W_{j}}(x_{j})}{\mathcal{J}_{\bar{W}_{j}}(\bar{x}_{j})}$

where
$W_{j}/\bar{W}_{j}=\tilde{F}_{\infty}^{j}W/\tilde{F}_{\infty}^{j}\bar{W}$ and
$x_{j}/\bar{x}_{j}=\tilde{F}_{\infty}^{j}x/\tilde{F}_{\infty}^{j}\bar{x}$.
As a result,

$\log\mathcal{J}h(x)=\sum_{j=0}^{\infty}\log\mathcal{J}_{W_{j}}(x_{j})-\log\mathcal{J}_{\bar{W}_{j}}(\bar{x}_{j}).$

Since $\psi^{\prime}\in[2k,2k+\frac{g}{2}]$, we obtain by (5)

$\displaystyle\quad\quad
2|\log\mathcal{J}_{W_{j}}(x_{j})-\log\mathcal{J}_{\bar{W}_{j}}(\bar{x}_{j})|$
$\displaystyle\leq\bigg{|}\log\left(1+4k_{j+1}^{2}+\psi_{j}^{\prime
2}\left(\frac{4}{g^{2}}+\left(1+\frac{4k_{j+1}}{g}\right)^{2}\right)+2\psi_{j}^{\prime}\left(\frac{2}{g}+2k_{j+1}\left(1+\frac{4k_{j+1}}{g}\right)\right)\right)$
$\displaystyle\quad-\log\left(1+4\bar{k}_{j+1}^{2}+\bar{\psi}_{j}^{\prime
2}\left(\frac{4}{g^{2}}+\left(1+\frac{4\bar{k}_{j+1}}{g}\right)^{2}\right)+2\bar{\psi}_{j}^{\prime}\left(\frac{2}{g}+2\bar{k}_{j+1}\left(1+\frac{4\bar{k}_{j+1}}{g}\right)\right)\right)\bigg{|}$
$\displaystyle\quad+|\log(1+\psi_{j}^{\prime 2})+\log(1+\bar{\psi}_{j}^{\prime
2})|$ $\displaystyle\leq
C\theta_{1}(|k_{j+1}-\bar{k}_{j+1}|+|\psi^{\prime}_{j}-\bar{\psi}^{\prime}_{j}|)+C\theta_{2}|\psi^{\prime}_{j}-\bar{\psi}^{\prime}_{j}|$
$\displaystyle\leq
C\theta_{1}(|t_{j+1}-\bar{t}_{j+1}|+|\psi^{\prime}_{j}-\bar{\psi}^{\prime}_{j}|)+C\theta_{2}|\psi^{\prime}_{j}-\bar{\psi}^{\prime}_{j}|$

where

$\theta_{1}^{-1/2}=1+4k_{\min}^{2}+4k_{\min}^{2}\left(\frac{4}{g^{2}}+\left(1+\frac{4k_{\min}}{g}\right)^{2}\right)+4k_{\min}\left(\frac{2}{g}+2k_{\min}\left(1+\frac{4k_{\min}}{g}\right)\right)>1$
$\theta_{2}^{-1/2}=1+4k_{\min}^{2}>1$

It also follows from (3) that

$\displaystyle|\psi^{\prime}_{j}-\bar{\psi}^{\prime}_{j}|$ $\displaystyle\leq
C|t_{j}-\bar{t}_{j}|+C\theta_{3}|\psi^{\prime}_{j-1}-\bar{\psi}^{\prime}_{j-1}|$
$\displaystyle\leq
C|t_{j}-\bar{t}_{j}|+C\theta_{3}|t_{j-1}-\bar{t}_{j-1}|+C\theta_{3}^{2}|\psi^{\prime}_{j-2}-\bar{\psi}^{\prime}_{j-2}|$
$\displaystyle\cdots$ $\displaystyle\leq
C|t_{j}-\bar{t}_{j}|+C\theta_{3}|t_{j-1}-\bar{t}_{j-1}|+\cdots+C\theta_{3}^{j-1}|t_{1}-\bar{t}_{1}|+C\theta_{3}^{j}|\psi^{\prime}_{0}-\bar{\psi}^{\prime}_{0}|$
$\displaystyle\leq
C\frac{|t_{0}-\bar{t}_{0}|}{\Lambda^{n}}+C\theta_{3}\frac{|t_{0}-\bar{t}_{0}|}{\Lambda^{j-1}}+\cdots+C\theta_{3}^{j-1}\frac{|t_{0}-\bar{t}_{0}|}{\Lambda}+C\theta_{3}^{j}|\psi^{\prime}_{0}-\bar{\psi}^{\prime}_{0}|$
$\displaystyle\leq
Cj\theta_{4}^{j}|t_{0}-\bar{t}_{0}|+C\theta_{3}^{j}|\psi^{\prime}_{0}-\bar{\psi}^{\prime}_{0}|$

where $\theta_{3}^{-1/2}=1+4k_{\min}/g>1$,
$\theta_{4}=\max\\{\theta_{3},\Lambda^{-1}\\}<1$ and $\Lambda$ is the
expansion rate of unstable curves.
Consequently,

(7)
$\displaystyle|\log\mathcal{J}_{W_{j}}(x_{j})-\log\mathcal{J}_{\bar{W}_{j}}(\bar{x}_{j})|\leq
Cj\Theta^{j}|t_{0}-\bar{t}_{0}|+C\Theta^{j}|\psi^{\prime}_{0}-\bar{\psi}^{\prime}_{0}|$

where $\Theta=\max\\{\theta_{1},\theta_{2},\theta_{3},\theta_{4}\\}<1$.

Finally we are ready to estimate the Jacobian. Observe that
$s_{+}(x,y)=s_{+}(\bar{x},\bar{y})$ since each pair $(x,\bar{x})$,
$(y,\bar{y})$ is connected by its corresponding stable manifold.

$\quad\quad|\log\mathcal{J}h(x)-\log\mathcal{J}(y)|$
$\leq\sum_{j=0}^{\infty}|\log\mathcal{J}_{W_{j}}(x_{j})-\log\mathcal{J}_{\bar{W}_{j}}(\bar{x}_{j})-\log\mathcal{J}_{W_{j}}(y_{j})+\log\mathcal{J}_{\bar{W}_{j}}(\bar{y}_{j})|$
$\leq\sum_{j<s_{+}(x,y)}\left(|\log\mathcal{J}_{W_{j}}(x_{j})-\log\mathcal{J}_{W_{j}}(y_{j})|+|\log\mathcal{J}_{\bar{W}_{j}}(\bar{x}_{j})-\log\mathcal{J}_{\bar{W}_{j}}(\bar{y}_{j})|\right)$
$\quad+\sum_{j\geq
s_{+}(x,y)}\left(|\log\mathcal{J}_{W_{j}}(x_{j})-\log\mathcal{J}_{\bar{W}_{j}}(\bar{x}_{j})|+|\log\mathcal{J}_{W_{j}}(y_{j})-\log\mathcal{J}_{\bar{W}_{j}}(\bar{y}_{j})|\right)$
$\leq
C\sum_{j<s_{+}(x,y)}(|x_{j}-\bar{x}_{j}|+|y_{j}-\bar{y}_{j}|)+C\sum_{j\geq
s_{+}(x,y)}j\Theta^{j}(|x_{0}-\bar{x}_{0}|+|y_{0}-\bar{y}_{0}|)$
$\quad+\Theta^{j}(|\psi^{\prime}(x_{0})-\bar{\psi}^{\prime}(\bar{x}_{0})|+|\psi^{\prime}(y_{0})-\bar{\psi}^{\prime}(\bar{y}_{0})|)$
$\leq
C\Lambda^{-s_{+}(x,y)}(|x_{s_{+}(x,y)}-\bar{x}_{s_{+}(x,y)}|+|y_{s_{+}(x,y)}-\bar{y}_{s_{+}(x,y)}|)$
$\quad+\Theta^{s_{+}(x,y)}(|x_{0}-\bar{x}_{0}|+|y_{0}-\bar{y}_{0}|+|\psi^{\prime}(x_{0})-\bar{\psi}^{\prime}(\bar{x}_{0})|+|\psi^{\prime}(y_{0})-\bar{\psi}^{\prime}(\bar{y}_{0})|)\leq
C\Theta^{s_{+}(x,y)}$

where the sum of small indices $j<s_{+}(x,y)$ is controlled by the distortion
estimate from Lemma 6.1 and the sum of large indices $j\geq s_{+}(x,y)$ is
controlled by (7). ∎

##  7\. Escaping and Bounded Orbits

Theorem 5 shows that the escaping orbits takes up a null set. However in this
section we show that the escaping orbits do exist and so do the bounded
orbits.

We introduce the notion of _proper standard pair_. A _standard pair_
$(W,\mu_{W})$ consists of an unstable curve $W$ and a _regular_ probability
measure $\mu_{W}$ supported on $W$, i.e. $\mu_{W}$ is absolutely continuous
and has a dynamically Hölder density. We say a standard pair is _proper_ if
there exists a large constant $C_{p}$ bounding the following the quantity

$\mathcal{Z}_{W}:=\sup_{\epsilon}\frac{\mu_{W}\\{r_{0}<\epsilon\\}}{\epsilon}.$

It is easy to see that in our case $\mu_{W}$ is the normalised Lebesgue
measure on the unstable curve and that $\mathcal{Z}_{W}=\frac{2}{|W|}$, so any
unstable curve $W$ longer than $\delta_{2}$ endowed with Lebesgue measure is a
proper standard pair. We also observe that $\delta_{2}$ can be made
arbitrarily small by choosing $C_{p}$ large. Therefore by Theorem 3 and Lemma
2.2, 2.3 in [13], we have the following central limit theorem for all proper
standard pairs:

######  Proposition 7.1.

There exists $\delta_{2}\gg 1$ such that on any unstable curve $W$ with
$|W|>\delta_{2}$ we have the following central limit theorem for dynamically
Hölder observables, i.e.

$\frac{1}{\sqrt{n}}\sum_{i=0}^{n-1}\varphi\circ\tilde{F}_{\infty}^{i}\stackrel{{\scriptstyle
dist}}{{\rightharpoonup}}\mathcal{N}(0,\sigma_{\varphi}^{2})$

where $\varphi$ is dynamically Hölder with zero average
$\int_{\mathbb{T}}\varphi d\tilde{\mu}=0$.

Now we prove Theorem 7.

###### Proof of Theorem 7.

Let us denote $(t_{n}^{\infty},v_{n}^{\infty})=F_{\infty}^{n}(t_{0},v_{0})$
and $(t_{n},v_{n})=F^{n}(t_{0},v_{0})$.
First we recall from Section 5 that the energy change $\gamma$ in the limit
map $F_{\infty}$ on the cylinder has zero average. Moreover, $\gamma$ is
dynamically Hölder as it is piecewise $C^{1}$ and its discontinuities are
located exactly on $\mathcal{S}^{+}$. Therefore by Proposition 7.1, $\exists
n_{0},A$ such that for every unstable curve $W$ longer than $\delta_{2}$

$\mathbb{P}_{W}\left(v_{n_{0}N_{0}}^{\infty}>v_{0}+A\sqrt{n_{0}N_{0}}\right)>\frac{1}{3}$

where $N_{0}$ is the constant from Lemma 6.3 $N_{0}$-Step Expansion.
By Lemma 6.5, if $\delta_{2}$ is sufficiently small, then for sufficiently
large $n_{0}$

$\mathbb{P}_{W}(r_{n_{0}N_{0}}<4\delta_{2})<\frac{1}{10}.$

We know from Section 3.2 that the limit map $F_{\infty}$ well approximates the
original collision map $F$ for large velocity with an error of order
$\mathcal{O}(v_{0}^{-1})$ on each continuity component of
$F_{\infty}^{n_{0}N_{0}}W$, thus we can choose $v_{*}\gg 1$ so large that if
$v_{0}>v_{*}$ everywhere on $W$, then we have

$\mathcal{P}_{W}(v_{n_{0}N_{0}}>v_{0}+A\sqrt{n_{0}N_{0}},r_{n_{0}N_{0}}>4\delta_{2})>\frac{1}{4}.$

By the estimate above, at least one component $W_{1}\subset
F_{\infty}^{n_{0}N_{0}}W$ contains a segment $\bar{W}_{1}$ longer than
$\delta_{2}$ and $v_{n_{0}N_{0}}>v_{0}+A\sqrt{n_{0}N_{0}}$ holds everywhere on
$\bar{W}_{1}$. By repeating the argument on $\bar{W}_{1}$, we get another
component $W_{2}\subset F_{\infty}^{n_{0}N_{0}}\bar{W}_{1}$ containing a
segment $\bar{W}_{2}$ longer than $\delta_{2}$ and the velocity increases by
another $A\sqrt{n_{0}N_{0}}$. Inductively, we construct an escaping orbit.
Similarly, we can construct an orbit whose velocity first increases by
$A\sqrt{n_{0}N_{0}}$ and then decreases by $A\sqrt{n_{0}N_{0}}$ and etc. In
that way we obtain an orbit whose energy remains in
$[v_{0}-2A\sqrt{n_{0}N_{0}},v_{0}+2A\sqrt{n_{0}N_{0}}]$. This holds for
arbitrarily large $v_{0}>v_{*}$, thus we have bounded orbits at arbitrarily
high energy level. ∎

##  8\. Conclusions

In this paper, we have studied a piecewise $C^{3}$-smooth Fermi-Ulam model in
a constant potential field. The collision map $F$ is well approximated by the
limit map $F_{\infty}$ for large velocities. $F_{\infty}$ covers a map
$\tilde{F}_{\infty}$ on a torus. For admissible wall motions we proved
ergodicity, exponential decay of correlations and central limit theorem for
dynamically Hölder observables. When our assumptions fail, there are
counterexamples in the class of analytic periodic platform motions by
Pustylnikov [24] where KAM islands exist for the limit map and the original
system possesses a positive measure set of escaping orbits. The ergodic and
statistical properties of the limit map $\tilde{F}_{\infty}$ established here
in turn imply that the escaping set has zero measure and the typical behavior
of the original collision dynamics $F$ is recurrent, but escaping and bounded
orbits still exist at arbitrarily high energy level.

It is also interesting to study long time evolution of the energy distribution
for typical high velocity trajectories, (cf. [5, 8] for similar results for
other systems). Besides, we note our results do not fully address the behavior
of low energy orbits. The problem becomes subtle when we come to the low
energy region as two consecutive collisions can happen in arbitrarily short
time interval, causing the system to be nonuniformly hyperbolic. In the
future, we hope to establish ergodicity for the the collision map $F$. This
would imply in particular that the almost every orbit is oscillatory, so that
the energy eventually comes close to any given value.

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11institutetext:  Institute for Big Data Analytics, Dalhousie University,
Halifax, Canada
22institutetext: Institute for Computer Science, Polish Academy of Sciences,
Warsaw, Poland
33institutetext: Memorial University of Newfoundland, St. John’s, Canada
44institutetext: Amirkabir University of Technology, Iran
44email<EMAIL_ADDRESS>
44email: {omid.amini<EMAIL_ADDRESS>
44email<EMAIL_ADDRESS>
44email<EMAIL_ADDRESS>
44email: {asayareh<EMAIL_ADDRESS>
# Observation Denoising in CYRUS Soccer Simulation 2D Team For RoboCup 2023

Aref Sayareh 33 Nader Zare 11 Omid Amini 11 Arad Firouzkouhi 44 Mahtab
Sarvmaili 11 Stan Matwin 1122 Amilcar Soares 33

###### Abstract

The RoboCup competitions hold various leagues, and the Soccer Simulation 2D
League is a major one among them. Soccer Simulation 2D (SS2D) match involves
two teams, including 11 players and a coach, competing against each other. The
players can only communicate with the Soccer Simulation Server during the
game. This paper presents the latest research of the CYRUS soccer simulation
2D team, the champion of RoboCup 2021. We will explain our denoising idea
powered by long short-term memory networks (LSTM) and deep neural networks
(DNN). The CYRUS team uses the CYRUS2D base code that was developed based on
the Helios and Gliders bases.

###### Keywords:

Robotic Denoising Soccer Simulation.

##  1 Introduction

RoboCup has hosted annual robotic soccer competitions since 1997, following
the proposal of robotic soccer games as a new research topic in 1992[1, 2, 3].
The primary objective of these competitions is to advance the robotics and
artificial intelligence field by implementing robotic soccer games. Several
leagues, including rescue, soccer simulation, and standard platform soccer,
have been created to achieve this.

The Soccer Simulation 2D (SS2D) league is one of the oldest RoboCup leagues.
In a soccer simulation 2D game, two teams containing 11 players and one coach
compete against each other. Each game takes 6000 cycles (10 minutes). In each
cycle, the players and coach receive observations from the RoboCup Soccer
Simulation Server and send actions based on those observations back to the
server. The server is responsible for controlling the game.

CYRUS has been a regular participant in RoboCup competitions since 2013 and
emerged as the champion of the SS2D league in RoboCup 2021. Over the years,
the CYRUS team has achieved first, second, second, third, fourth, and fifth
places in RoboCup in 2021, 2022, 2018, 2019, 2017, and 2014, respectively.
CYRUS also achieved first place in IranOpen in 2018 and 2014, RoboCup Asia-
Pacific in 2018, and second place in JapanOpen in 2020.

In this paper, we will first discuss related research by other teams, focusing
on topics such as AI and robotics in soccer simulation 2D. We will then
present our previous works, including our released applications and codes.
Next, we will explore the issue of noise in soccer simulation 2D, presenting
our ideas for addressing this challenge. Finally, we will outline our plans
for future work in this area.

###  1.1 Related Work

In this part, we will mention some latest novel ideas presented by SS2D teams
in recent years.

YuShan2022 adopted an offline training model and LambdaMART algorithm to
enhance their team’s offensive ability through movement direction
selection[4]. RobôCIn researched an adaptive strategy based on time, score,
and the opponent team’s pressure. Also, they developed and released a log
analyzer software[5]. Persepolis team improved main behaviours such as pass,
shoot, dribbling, and marking; also used a ranking algorithm to improve chain-
action performance[6]. Persepolis also implemented a Path Planning algorithm
to improve its offensive strategy[7, 8]. Helios2022 conducted numerical
experiments to investigate the effect of using Anonymous mode on game
results[9]. HfutEngine2D used genetic algorithms to adjust parameters and
attempted to use neural networks for rational decision-making, significantly
improving the team’s strength during recent tests against other RoboCup
teams.[10]. Alice2D’s work centered around optimizing defense strategy using
the Monte-Carlo tree and reinforcement learning algorithms.[11]. Hades2D team
improved individual players’ offensive and defensive skills, significantly
decreasing the average number of conceded goals against the base.[12, 13].
FRA-UNIted utilized reinforcement learning (RL) techniques to improve their
ability to intercept the ball from a player in possession. They created a
Markov decision process (MDP) environment with two players: the agent learning
to intercept and the dribbling player. Through this environment, the agent
learned and improved its intercepting skills over time.[14].

###  1.2 CYRUS Previous Works

In this section, we briefly explain the history of our research. CYRUS was
established in 2013 and has participated in many competitions since then.
CYRUS Team researched various aspects of the game, starting with passing
behavior, unmarking strategy, and formation[15, 16]. After that, we shifted
our attention to defensive behaviour[17], shooting algorithms, and
communication modules[18]. In 2017, the CYRUS team developed the opponent
behavior prediction module[19, 20]. In recent years, CYRUS Team has mainly
been developing the pass prediction and unmarking module, and Data Generator
framework [21, 22, 23, 24, 25]. Some of our research work, including Cyrus2D
Base code, CYRUS 2014 Source Code, PYRUS - Python SS2D Base Code, and SS2D Log
Analyzer, has been released in recent years.

####  1.2.1 Cyrus 2014

Cyrus team released its source code used in the 2014 RoboCup competition,
where they achieved fifth place. The source code can be an excellent resource
for other teams and new teams interested in participating in the RoboCup
competition. The code provides valuable insights into the team’s approach,
strategies, and implementations that can be helpful for others to improve
their performance. The code is available on our GitHub111Cyrus 2014 Source
https://github.com/naderzare/cyrus2014..

####  1.2.2 Cyrus2D Base

Cyrus2D Base[26] code is a soccer simulation base code created by combining
Helios Base[27], Gliders base[28], and the best features of the Cyrus team,
which won first place in the RoboCup Soccer Simulation 2D league in 2021. This
base code is an excellent starting point for new teams just starting to work
on their soccer simulation projects. The code provides a solid foundation for
building the functionalities required for the RoboCup competition. The code is
available on our GitHub222Cyrus Base Source
https://github.com/Cyrus2D/Cyrus2DBase.

####  1.2.3 PYRUS Base

Pyrus Base code is a soccer simulation base code developed in Python,
providing an excellent platform for researchers to develop and test their
Machine Learning algorithms. This base code benefits teams interested in
applying artificial intelligence techniques to enhance performance. The code
is available on our GitHub333Pyrus Base https://github.com/Cyrus2D/Pyrus-
SS2D-Base.

####  1.2.4 Cyrus AutoTest2D

This software was initially developed by the WrightEagle team[29], and Cyrus
team members upgraded and simplified it. The software helps teams run many
parallel games to test their performance. Cyrus AutoTest provides a systematic
approach to evaluating teams’ performance in the soccer simulation league, and
the software can help teams identify their strengths and weaknesses to improve
their performance. The code is available on our GitHub444AutoTest
https://github.com/Cyrus2D/AutoTest2D.

####  1.2.5 AutoTune2D

Cyrus team developed this tool that allows researchers to fine-tune their
parameters alongside AutoTest. AutoTune can be a valuable tool for researchers
who want to optimize their team’s performance by adjusting various parameters
to find the best combination. The code is available on our GitHub555AutoTune
https://github.com/Cyrus2D/AutoTune2D.

####  1.2.6 Cyrus Log analyzer

This software can analyze soccer simulation 2D game logs to find statistical
values, including the number of passes, dribbles, and other relevant
statistics. This software can provide valuable insights into a team’s
performance, helping teams identify areas of improvement and fine-tune their
strategies. The Cyrus Log analyzer can help teams optimize their performance
by providing detailed information about their gameplay. The code is available
on our GitHub666CyrusLog Analayzerhttps://github.com/Cyrus2D/CyrusLogAnalyzer.

##  2 Denoising

###  2.1 Observation Noise In SS2D

In the Soccer Simulation 2D league, accurately calculating the position of
each object, including players and the ball, presents a significant challenge
due to agents’ visual sensor limitations.

In this multi-agent environment, the agent can use three different vision
angles: 180(Wide), 120(Normal), and 60(Narrow) degrees. However, using each
angle comes at a cost. For instance, if the agent chooses to use the Wide
angle, it must wait three cycles before using the sensor again. Similarly, for
the Normal angle and Narrow angle, it should wait for 2 and 1 cycle,
respectively. Choosing the right vision angle is important to balance
performance and reduce the delay caused by limited sensor usage.

To address this issue, a parameter called ’pos_count’ is defined for each
object, which indicates the number of cycles that have elapsed since the last
time the sensor received the object’s position.

In addition, the relative distance and angle information that the agent
receives from the visual sensor are quantized, resulting in more noise in the
distance measurement when the object is farther away. Furthermore, the
floating-point part of the angle measurement is rounded.

###  2.2 Helios Base Denoising Method

HeliosBase applies a technique to tackle the quantized noise by considering
the object’s distance. Specifically, it calculates the average range of
distances at which the object can be detected. This method is also applied to
the relative angle between the player and the object to refine the estimation
further. After these initial estimates, the method can determine the object’s
relative position before calculating its global position. This approach allows
for improved accuracy in object localization, which is crucial in many
applications such as robotics and computer vision.

###  2.3 Denoising by Machine Learning.

Our proposed approach aims to reduce the noise in the data when the object’s
pos_count is more than a cycle. To achieve this, we plan to investigate
various machine learning techniques, including Deep Neural Networks (DNNs)[30]
and Long Short-Term Memory (LSTM)[31] networks. These models’ predicted
positions are closer to the true objects’ than the last seen positions. The
time series models will be trained on input sequences that include the
position and sensor data from the last five cycles. We aim to find the best
machine learning model to predict the opponents’ position.

###  2.4 Generating Data

In order to gather the necessary data for our study, we ran a total of 400
games between Cyrus2DBase and HeliosBase. The collected data includes the
positional and velocity information of the ball and 22 players within the game
and the direction of the players’ bodies. The data also includes the pos_count
parameter for each player. The position data of each object consists of both
noisy and original values.

##  3 Results

In addition to training the DNN and LSTM models, we also experimented with
different sizes and configurations of these models to find the optimal
architecture for denoising the positional data. After extensive testing, we
found the LSTM model with 512 and 256 neurons and all ReLU activations in the
time series model and the DNN model with 512, 256, 128, 64, and 32 neurons all
ReLU activations achieved the best performance. Furthermore, we discovered
that the LSTM model was more effective in predicting the accurate position of
the players than the DNN model. Therefore, we focused on improving our
solution’s DNN and LSTM models.

Figure 1: Comparison of DNN accuracy and last-seen position accuracy. The
green color shows a better performance with DNN, whereas the blue color
indicates better performance in the last seen position. Insufficient data to
compare these two methods are shown in black color.
Figure 2: Comparison of LSTM accuracy and last-seen position accuracy. The red
color shows a better performance with LSTM, whereas the blue color indicates
better performance in the last seen position. Insufficient data to compare
these two methods are shown in black color.
Figure 3: Comparison of LSTM accuracy and DNN accuracy. The RED color shows a
better performance with LSTM, whereas the green color indicates better
performance in DNN. Insufficient data to compare these two methods are shown
in black color.

Figures 1, 2 and 3 presented in the heatmap compare errors among three
methods: DNN, LSTM, and Agent Vision data. The heatmaps are based on the
number of cycles an agent has not seen the player and the distance between the
agent and the player. The agent represented in the figures is number 9 of the
left team, and the object being observed is player 5 of the left team. LSTM
has a higher level of accuracy than DNN, while DNN is more precise than player
vision. Therefore, LSTM performs more accurately than the other two methods
for this specific task.

##  4 Future work

There are several challenges associated with the proposed models. Firstly, the
accuracy of the models may decrease if a team’s strategy and formation change,
as the models primarily learn from the team’s historical data. Secondly, there
may not be existing models for new teams in the league due to the lack of
data. In such cases, one approach could be choosing a pre-trained model for
other teams similar to the new ones to predict their performance. Another
approach could be implementing online learning techniques that learn the
opponent’s strategy during a game, which can be challenging and require
further investigation.

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