# A Rectangular Billiard with Moving Slits

Jing Zhou

###### Abstract.

We describe an exponential Fermi accelerator in a two-dimensional billiard
with a moving slit. We have found a mechanism of trapping regions which
provides the exponential acceleration for almost all initial conditions with
sufficiently high initial energy. Under an additional hyperbolicity
assumption, we estimate the waiting time after which most high-energy orbits
start to gain energy exponentially fast.

The author thanks Vered Rom-Kedar for posing this problem and for many useful
discussions on this subject. The author is also much obliged to Dmitry
Dolgopyat for enlightening comments and advice. The research was partially
supported by NSF grant DMS 1665046 and by BSF grant 2016105. Part of this work
was done at Weizmann Institute, where the hospitality and excellent working
conditions are gratefully acknowledged.

##  1\. Introduction

In an attempt to explain the existence of high energy particles in cosmic
rays, Fermi [8] in 1949 proposed a model in which charged particles undergo
repeated reflections in moving magnetic fields. Later in 1961 Ulam [26]
proposed that a similar mechanism should appear in finite degree of freedom
systems. He described a toy model where a particle bounces elastically between
two walls, one fixed and the other moving periodically. Ulam [26] performed
numerical experiments on a piecewise linear model and conjectured that there
exist escaping orbits whose energy tend to infinity with time. Since then
extensive efforts have been made by both mathematicians and physicists to
locate escaping orbits in various settings (see [18, 6, 10] for surveys on
this subject).

Notably the KAM theory has eliminated the possibility of such escaping orbits
in one-dimensional Fermi-Ulam model for sufficiently smooth wall motions [16]
[23] [24], as the prevalence of invariant curves forces all orbits to be
bounded. However, unbounded solution can still be obtained in nonsmooth case.
For example, Zharnitsky [27] has found a type of unbounded orbits in the
Ulam’s piecewise linear case, which grows linearly with time. In a one-
dimensional Fermi-Ulam model with one discontinuity, De Simoi and Dolgopyat
[3] has showed that there exists a parameter that completely shapes the large
energy behavior of the system: in the hyperbolic regime the escaping set has
zero measure but full Hausdorff dimension while in some elliptic cases the
escaping orbits have infinite measure. There are many interesting question
pertaining to Fermi-Ulam model with non-periodic wall motion. We refer the
reader to [28, 15] for the results in quasi-periodic case. One and a half
degree of freedom models where the motion between collision is not free but is
subjected to a potential are discussed, for example in [22, 20, 21, 5, 2, 1].

Two-dimensional moving billiard models are natural generalization of the one-
dimensional Fermi acceleration model and can often provide chaotic orbits even
in the smooth case. For example, unbounded orbits were found in billiard
models with the smoothly breathing boundary [13, 14, 17]. In a Lorentz gas
model [19], the average velocity of particles grows linearly in time for
stochastic perturbation of scatterer boundaries and quadratically for periodic
perturbation. Exponential growing orbits for non-autonomous billiards were
constructed in [12] but in general it remains challenging to detect a positive
measure set of exponentially growing orbits. Exponential acceleration is also
conjectured to be generic for oscillating mushrooms [11]. The models which are
closest to our setting are the following. Shah, Turaev and Rom-Kedar [25]
investigated a rectangular billiard model with a moving slit. They proposed a
random process approximation for the particle energy from which the calculated
exponential growth rate agrees well with the numerical results for typical
trajectories in the non-resonant case. They also observed numerically in [25]
that the growth rate in the resonant case is significantly higher than that in
the non-resonant case. Later they generalized the idea to higher-dimensional
time-dependent billiards and obtained a new class of numerically robust
exponential accelerators [9]. The model presented in this paper was inspired
by their work.

We study a rectangular billiard with two moving slits. In our model, the
billiard table is a unit square. Two slits are moving vertically in the table
with the length of left slit $\lambda$ and the length of the right slit
$1-\lambda$, where $\lambda\in(0,1)$ is a constant. The motion of the two
slits are described by two $C^{2}$ 2-periodic functions $f_{L}(t)$ and
$f_{R}(t)$ respectively. A massless ball bounces elastically against the
moving slits as well as the boundary of the rectangular table.

01$\lambda$1$f_{L}(t)$$f_{R}(t)$ Figure 1. Rectangular Billiard with Moving
Slits

In this paper we study the simplest resonant case. Namely, we assume that the
horizontal speed the ball is 1, so the horizontal coordinate of the ball is
periodic with period 2. Hence we have 1:1 resonance between the period of
moving slits and the period the horizontal motion of the ball. The ball
experiences two jumps between the left and the right parts of the table during
each period. We denote by $x_{0}$ the starting horizontal position of the
ball. We assume without loss of generality that the ball starts from the left
part, i.e. $0\leq x_{0}<\lambda$. The ball jumps from the left slit to right
one at time $t_{1}^{*}=\lambda-x_{0}$ and then from right to the left at time
$t_{2}^{*}=2-\lambda-x_{0}$.

We record the time and the vertical velocity of the ball immediately after
each collision with the slits. We exclude from our discussion the trajectories
having a collision at $x=\lambda.$ The excluded orbits constitute a measure
zero set among all the initial conditions.

We describe in this paper a new exponential accelerator. We show that almost
all initial conditions with sufficiently high initial velocity produce
exponential energy growth in the future, provided that the relative positions
of the two slits change at the time of the two jumps between left and right
parts of the table. Moreover, under an additional hyperbolicity assumption we
estimate the waiting time after which most high-energy orbits start to
accelerate exponentially.

##  2\. Main Results

In this section we describe the exponential accelerator we have found.
For a wide range of choices in $\lambda$ and $x_{0}$, the relative positions
of the left and right slits change when the ball jumps from one slit to the
other at time $t_{1}^{*}$ and $t_{2}^{*}$ (c.f. Figure 2). A trapping region
is created in this case and the ball starts to gain energy exponentially fast
once it gets trapped.

######  Theorem 1.

Assume that $f_{L}(t_{1}^{*})<f_{R}(t_{1}^{*})$ and
$f_{L}(t_{2}^{*})>f_{R}(t_{2}^{*})$ or $f_{L}(t_{1}^{*})>f_{R}(t_{1}^{*})$ and
$f_{L}(t_{2}^{*})<f_{R}(t_{2}^{*})$. Then there exists $V_{*}\gg 1$, which
depends only on $f_{L}$ and $f_{R}$, such that almost every orbit whose
initial speed is greater than $V^{*}$ eventually gains energy exponentially in
time. In particular, the set of initial conditions $(t_{0},v_{0})$ which do
not enjoy exponential energy growth has finite measure.

In the presence of a trapping region, with additional hyperbolicity
assumptions we estimate the waiting time after which most high-energy orbits
start to accelerate exponentially.
We define a new function $f$ as follows

$f(t)=\begin{cases}f_{L}(t)&0<t<t_{1}^{*}\hbox{ or }t_{2}^{*}<t<2\\\
f_{R}(t)&t_{1}^{*}<t<t_{2}^{*}.\end{cases}$

We introduce a new quantity $Tr$. If the lower chamber is trapping, then we
define in the upper chamber

$\displaystyle Tr^{U}$
$\displaystyle=\left(\frac{1-f_{1}^{-}}{1-f_{1}^{+}}-a_{1}\beta\right)\left(\frac{1-f_{2}^{-}}{1-f_{2}^{+}}-a_{2}\alpha\right)+\left(\frac{1-f_{1}^{+}}{1-f_{1}^{-}}-a_{1}\alpha\right)\left(\frac{1-f_{2}^{+}}{1-f_{2}^{-}}-a_{2}\beta\right)$
$\displaystyle\ \ \ \ \ -a_{1}a_{2}\alpha\beta$

where $f_{i}^{\pm}=f(t_{i}^{*}\pm),$
$a_{i}=\dot{f}_{i}^{-}(1-f_{i}^{+})-\dot{f}_{i}^{+}(1-f_{i}^{-}),$

$\alpha=\left(\int_{0}^{t_{1}^{*}}+\int_{t_{2}^{*}}^{2}\right)\frac{ds}{(1-f(s))^{2}}\quad\text{and}\quad\beta=\int_{t_{1}^{*}}^{t_{2}^{*}}\frac{ds}{(1-f(s))^{2}}.$

If the upper chamber is trapping, then we define in the lower chamber

$Tr^{L}=\left(\frac{f_{1}^{-}}{f_{1}^{+}}-a_{1}^{\prime}\beta^{\prime}\right)\left(\frac{f_{2}^{-}}{f_{2}^{+}}-a_{2}^{\prime}\alpha^{\prime}\right)+\left(\frac{f_{1}^{+}}{f_{1}^{-}}-a_{1}^{\prime}\alpha^{\prime}\right)\left(\frac{f_{2}^{+}}{f_{2}^{-}}-a_{2}^{\prime}\beta^{\prime}\right)-a_{1}^{\prime}a_{2}^{\prime}\alpha^{\prime}\beta^{\prime}$

where $a_{i}^{\prime}=\dot{f}_{i}^{+}f_{i}^{-}-\dot{f}_{i}^{-}f_{i}^{+},$
$\alpha^{\prime}=\left(\int_{0}^{t_{1}^{*}}+\int_{t_{2}^{*}}^{2}\right)\frac{ds}{f(s)^{2}}$
and $\beta^{\prime}=\int_{t_{1}^{*}}^{t_{2}^{*}}\frac{ds}{f(s)^{2}}$.

######  Theorem 2.

Assume $\lambda$ and $x_{0}$ are such that the relative positions of two slits
change at two critical jumps and that $|Tr|>2$. Then there is $K,\zeta>0$ such
that for any $\epsilon>0$, there exists $V_{0}=V_{0}(\epsilon)$ and
$T=T(\epsilon)$ such that for each $V\geq V_{0}$ the complement of set

$\left\\{(t_{0},v_{0}):|v_{0}|\in[V,V+1]:\forall t\geq
T\quad|v(t)|\geq\frac{|v(T)|}{K}e^{\zeta t}\;\right\\}$

has measure less than $\epsilon$, i.e. most orbits with initial energy
$|v_{0}|>V_{0}$ start to accelerate exponentially after time $T$.

The proof of this result is constructive. In particular, $T$ depends
logarithmically on $\epsilon$ (see equations (12) and (13)).

Figure 2. Trapping Regions

The plot of trapping regions for $f_{L}(t)=0.3\cos(\pi t)+0.5$,
$f_{R}(t)=0.3\sin(\pi t)+0.5$. The blue part indicates the values of
$\lambda,x_{0}$ for which a trapping region exists: the upper rectangle is
where the upper chamber is trapping and the lower triangle is where the lower
chamber is trapping. The shaded part displays where the hyperbolicity
assumption $|Tr|>2$ holds.

######  Example 2.1.

To illustrate our results, we consider the case where

(1) $f_{L}(t)=0.3\cos(\pi t)+0.5,\quad f_{R}(t)=0.3\sin(\pi t)+0.5.$

Then

$\Delta(t):=f_{L}(t)-f_{R}(t)=0.3\sqrt{2}\cos\left(\pi
t+\frac{\pi}{4}\right).$

A trapping region exists for $\lambda,x_{0}$ such that either

$\Delta(\lambda-x_{0})>0,\quad\Delta(2-\lambda-
x_{0})<0\quad\text{or}\quad\Delta(\lambda-x_{0})<0,\quad\Delta(2-\lambda-
x_{0})>0.$

The former case is equivalent to $\lambda-x_{0}<0.25,\
0.75<\lambda+x_{0}<1.75$; the upper chamber is trapping and the hyperbolicity
assumption holds if $|Tr^{L}|>2$.
The latter case is equivalent to $\lambda-x_{0}>0.25,\ \lambda+x_{0}<0.75$;
the lower chamber is trapping and the hyperbolicity assumption holds if
$|Tr^{U}|>2$.
Figure 2 demonstrates that for $f_{L},f_{R}$ defined above by (1) the
assumptions of Theorems 1 and 2 hold for a sizable set of parameters.

The structure of the rest of the paper is the following. In Section 3 we
describe the collision map. In Section 4 we derive the normal form for the map
obtained by considering the next collision with the moving wall after the ball
switches from left to right chamber or vice versa. The proof of Theorem 1 is
given in Section 5 and the proof of Theorem 2 is given in Section 6. In
Section 7 we summarize the tools developed in the present paper and discuss
open problems.

##  3\. Preliminaries

Since the horizontal speed of the ball stays constant, only the vertical speed
contributes to the energy change of the ball. This is why we only need to
record the time $t$ and the vertical velocity $v$ immediately after each
collision. Let us denote by $F$ the collision map.

For $i=1,2$, we denote as $R_{i}$ the strip in the $(t,v)$-plane bounded by
the singularity line $\mathcal{S}_{i}=\\{t=t_{i}^{*}\\}$ and its image
$F\mathcal{S}_{i}.$ Also let $\tilde{R}_{i}=F^{-1}R_{i}$. We subdivide the
singular strips $R$ into upper and lower chamber parts $R^{+}$ and $R^{-}$.

There are four possible scenarios when the ball makes a jump: the ball always
hits the slits from above or below, the ball first hits from above then from
below and vice versa.

We start with the easiest case when the ball always stays in the same chamber.
Then the system is effectively equivalent to a Fermi-Ulam model with the
motion (height) of the wall being the piecewise smooth 2-periodic function
$f(t)$ with two jump discontinuities at $t_{1}^{*}$ and $t_{2}^{*}$.

Suppose that the ball is initially in the upper chamber. We omit the subscript
$i$ as the formulas for passing through the two singularities are the same. If
for $(t,v)\in\tilde{R}$ we have $f(t^{*}+)-f(t)<v(t^{*}-t)<2-f(t)-f(t^{*})$,
then the ball ends in the upper chamber after jumping and the model is
equivalent to the one with a fixed ceiling and a moving floor (c.f. Figure 3
on the left).
Two consecutive collisions $(t_{n},v_{n})$ and
$(t_{n+1},v_{n+1})=F(t_{n},v_{n})$ satisfy

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

Figure 3. Equivalent Fermi-Ulam Models for the Upper/Lower Chambers

Similarly, suppose that the ball is initially in the lower chamber. If for
$(t,v)\in\tilde{R}$ we have $f(t)-f(t^{*}+)<-v(t^{*}-t)<f(t)+f(t^{*})$, then
the ball ends in the lower chamber after jumping and the model is equivalent
to the one with a moving ceiling and a fixed floor (c.f. Figure 3 on the
right). Two consecutive collisions satisfy

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

Now let us examine the switching cases.

Suppose that the ball is initially in the upper chamber and two consecutive
collisions still follow Equation 2 before the ball jumps from one slit to the
other. However, when the ball jumps, if the next chamber is above the previous
one when the ball passes through the singularities, then there is a
possibility that the ball enters the lower chamber. More precisely, for
$(t,v)\in\tilde{R}$, if $v(t_{*}-t)>2-f(t)-f(t_{*}+)$, then the ball collides
with the ceiling and then enters the lower chamber (c.f. Figure 4 on the
left); while if $v(t_{*}-t)<f(t_{*}+)-f(t)$, then the ball enters the lower
chamber immediately after it leaves the previous slit (c.f. Figure 4 on the
right).

Figure 4. From Upper to Lower Cases

Two consecutive collisions satisfy the following Equation 4 in the first case

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

and the following Equation 5 in the second case

(5) $\left\\{\begin{aligned} &v_{n}(t_{n+1}-t)=f(t_{n+1})-f(t_{n})\\\
&v_{n+1}=-v_{n}+2\dot{f}(t_{n+1}).\end{aligned}\right.$

On the other hand, suppose the ball is initially in the lower chamber and two
consecutive collisions still follow Equation 3 before the ball jumps from one
slit to the other. When the ball jumps, if the next chamber is below the
previous one when the ball passes through the singularities, then there is a
possibility that the ball enters the upper chamber. More precisely, for
$(t,v)\in\tilde{R}$, if $-v(t_{*}-t)>f(t)+f(t_{*}-)$, then the ball collides
with the floor and enters the upper chamber (c.f. Figure 5 on the left); while
if $-v(t_{*}-t)<f(t)-f(t_{*}-)$, then the ball enters the lower chamber
immediately after it leaves the previous slit (c.f. Figure 5 on the right).

Figure 5. From Lower to Upper Cases

Two consecutive collisions satisfy the following Equation 6 in the first case

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

and the following Equation 7 in the second case

(7) $\left\\{\begin{aligned} &v_{n}(t_{n+1}-t)=f(t_{n+1})-f(t_{n})\\\
&v_{n+1}=-v_{n}+2\dot{f}(t_{n+1}).\end{aligned}\right.$

##  4\. The Normal Form

In this section we study how the (vertical) velocity of the ball changes after
one period $\Delta t=2$ given sufficiently large initial energy. We will first
approximate the collision map $F$ with an action-angle coordinate away from
singularities. Then we examine the collision dynamics when the ball passes
through singularities.

###  4.1. The Action-Angle Coordinate

First we suppose that the ball collides with the slit from above and it does
not make a jump at nearby collisions.
Let us denote $l(t)=1-f(t)$ and $\mathcal{L}_{*}=\int_{0}^{2}l^{-2}(s)ds$.

######  Lemma 4.1.

For $(t,v)\notin R_{i}\cup\tilde{R}_{i}$ $(i=1,2)$ and $v\gg 1$, there exists
an action-angle coordinate
$(\theta,I)=\Psi_{U}(t,v)\in\mathbb{R}/2\mathbb{Z}\times\mathbb{R}_{+}$ such
that

$\theta_{n+1}=\theta_{n}+\frac{2}{I_{n}}+\mathcal{O}\left(\frac{1}{I_{n}^{4}}\right),\
I_{n+1}=I_{n}+\mathcal{O}\left(\frac{1}{I_{n}^{3}}\right).$

In fact,
$\displaystyle\theta=\theta(t)=\frac{2}{\mathcal{L}_{*}}\int_{0}^{t}\frac{ds}{l(s)^{2}}\mod
2$, $\displaystyle
I=I(t,v)=\frac{\mathcal{L}_{*}}{2}\left(lv+l\dot{l}+\frac{l^{2}\ddot{l}}{3v}\right)$.

###### Proof.

We can check the formula by a direct computation (c.f. Lemma 2.2 in [3]), or
we can derive it in an inductive way (c.f. Section 2.2 in [4]). The basic idea
is to find higher-order adiabatic invariants. For example, observe that

$v_{n+1}-v_{n}\approx-2\dot{l}(t_{n}),\ \
t_{n+1}-t_{n}\approx\frac{2l(t_{n})}{v_{n}}.$

This leads to the Euler scheme of the following ODE

$\frac{dv}{dt}=\frac{-v\dot{l}}{l}$

which in turn gives us the zeroth order adiabatic invariant $I=lv$. Then we
update the scheme by replacing $v$ with $I$ and look for the first order
adiabatic invariant, etc. This scheme terminates at the second order adiabatic
invariant $\displaystyle I=lv+l\dot{l}+\frac{l^{2}\ddot{l}}{3v}$.
Next, the formula for $\theta$ can be obtained reversely by solving the ODE

${\theta}^{\prime}\frac{2l}{v}=\frac{2}{lv}$

which leads to $\displaystyle\theta(t)=\int_{0}^{t}l^{-2}(s)ds$.
We observe that only the order $v$ term in $I$ is used to derive the formula
for $\theta$ and it seems to produce an estimate only up to first order

$\theta_{n+1}-\theta_{n}=2/I_{n}+\mathcal{O}(I_{n}^{-2}).$

But in fact by noting the Taylor expansion of $l^{-2}$ and that

$t_{n+1}-t_{n}=\frac{2l(t_{n})}{v_{n}}+\frac{2l(t_{n})\dot{l}(t_{n})}{v_{n}^{2}}+\frac{2l(t_{n})\dot{l}(t_{n})^{2}+2l(t_{n})^{2}\ddot{l}(t_{n})}{v_{n}^{3}}+\mathcal{O}(v_{n}^{-4})$

we obtain that

$\displaystyle\int_{t_{n}}^{t_{n+1}}l^{-2}(s)ds$
$\displaystyle=\frac{t_{n+1}-t_{n}}{l_{n}^{2}}-\frac{\dot{l}_{n}}{l_{n}^{3}}(t_{n+1}-t_{n})^{2}+(\frac{\dot{l}_{n}^{2}}{l_{n}^{4}}-\frac{\ddot{l}_{n}}{3l_{n}^{3}})(t_{n+1}-t_{n})^{3}+\mathcal{O}(v_{n}^{-4})$
$\displaystyle=\frac{2}{l_{n}v_{n}}-\frac{2\dot{l}_{n}}{t_{n}v_{n}^{2}}+\frac{2\dot{l}_{n}^{2}-\frac{2}{3}l_{n}\ddot{l}_{n}}{l_{n}v_{n}^{3}}+\mathcal{O}(v_{n}^{-4})$
$\displaystyle=\frac{2}{I_{n}}+\frac{2\dot{l}_{n}}{v_{n}I_{n}}+\frac{2l_{n}\ddot{l}_{n}}{3v_{n}^{2}I_{n}}-\frac{2\dot{l}_{n}}{l_{n}v_{n}^{2}}+\frac{2\dot{l}_{n}^{2}-\frac{2}{3}l_{n}\ddot{l}_{n}}{l_{n}v_{n}^{3}}+\mathcal{O}(v_{n}^{-4})$
$\displaystyle=\frac{2}{I_{n}}+\frac{2\dot{l}_{n}^{2}}{v_{n}^{2}I_{n}}+\frac{2l_{n}\ddot{l}_{n}}{3v_{n}^{2}I_{n}}+\frac{2\dot{l}_{n}^{2}-\frac{2}{3}l_{n}\ddot{l}_{n}}{l_{n}v_{n}^{3}}+\mathcal{O}(v_{n}^{-4})$
$\displaystyle=\frac{2}{I_{n}}+\mathcal{O}(v_{n}^{-4})$

where $l_{n}=l(t_{n})$ and $\dot{l}_{n}=\dot{l}(t_{n})$, which produces the
desired third order estimate.

Finally, we need to rescale $\theta$ (and hence $I$) to make $\theta$ 2
periodic. ∎

Next we assume that the ball collides at the slits from below and it does not
make a jump at nearby collisions.
We introduce a new function $g(t)=f(t)+1$. Then Equation 3 becomes the same as
Equation 2 with $g$ in place of $f$

(8) $\left\\{\begin{aligned} &v_{n+1}=v_{n}+2\dot{g}(t_{n+1})\\\
&2-g(t_{n})-g(t_{n+1})=v_{n}(t_{n+1}-t_{n})\end{aligned}\right.$

Therefore all the computation above in Lemma 4.1 applies with $g$ in the place
of $f$.
We define $m(t)=1-g(t)=-f(t)$ and $\mathcal{M}_{*}=\int_{0}^{2}m(s)^{-2}ds$.
We have an action-angle coordinate if the collision occurs in the lower
chamber away from singularities

######  Lemma 4.2.

For $(t,v)\notin R_{i}\cup\tilde{R}_{i}$ $(i=1,2)$ and $v\ll-1$, there exists
an action-angle coordinate
$(\zeta,J)=\Psi_{L}(t,v)\in\mathbb{R}/2\mathbb{Z}\times\mathbb{R}_{+}$ such
that

$\zeta_{n+1}=\zeta_{n}+\frac{2}{J_{n}}+\mathcal{O}\left(\frac{1}{J_{n}^{4}}\right),\
J_{n+1}=J_{n}+\mathcal{O}\left(\frac{1}{J_{n}^{3}}\right).$

In fact,
$\displaystyle\zeta=\zeta(t)=\frac{2}{\mathcal{M}_{*}}\int_{0}^{t}\frac{ds}{m(s)^{2}}\mod
2$, $\displaystyle
J=J(t,v)=\frac{\mathcal{M}_{*}}{2}\left(mv+m\dot{m}+\frac{m^{2}\ddot{m}}{3v}\right)$.

###  4.2. The Normal Forms

In this section we present the Poincaré map $P$ from one singular strip to the
other in four possible scenarios. We assume the initial energy of the ball is
sufficiently large $|v_{0}|>V_{*}$ for some large $V_{*}$ in all the cases.

####  4.2.1. The Upper-Upper Chamber Case

We begin with the upper-upper chamber case, i.e. the ball stays in the upper
chamber both before and after it makes a jump. Lemma 4.1 already depicts the
dynamics away from singularities. Now let us scrutinize what occurs near the
singularities $t_{i}^{*}$ ($i=1,2$) when the ball makes a jump.

$t$$v$$\tilde{R}_{1}^{+}$$R_{1}^{+}$$\tilde{R}_{2}^{+}$$R_{2}^{+}$$t_{1}^{*}$$t_{2}^{*}$$V$$(t_{1},v_{1})$$(\tilde{t}_{2},\tilde{v}_{2})$$(\bar{t}_{2},\bar{v}_{2})$$F^{n_{1}}$
Figure 6. The Poincaré Map $P^{12}_{UU}$ on the Singular Strips

For $(t_{1},v_{1})\in R_{1}^{+}$ with $v_{1}>V_{*}$, we denote
$(\tilde{t}_{2},\tilde{v}_{2})=F^{n_{1}}(t_{1},v_{1})\in\tilde{R}_{2}^{+}$,
where $\displaystyle n_{1}=[\frac{I_{1}}{2}(\theta_{2}^{*}-\theta_{1})]$ and
$\displaystyle\theta_{2}^{*}=\frac{2}{\mathcal{L}_{*}}\int_{0}^{t_{2}^{*}}\frac{ds}{l(s)^{2}}$,
and $(\bar{t}_{2},\bar{v}_{2})=F(\tilde{t}_{2},\tilde{v}_{2})\in R_{2}^{+}$.
Similarly, for $(t_{2},v_{2})\in R_{2}^{+}$ with $v_{2}\gg 1$, we denote
$(\tilde{t}_{1},\tilde{v}_{1})=F^{n_{2}}(t_{2},v_{2})\in\tilde{R}_{1}^{+}$,
where $\displaystyle n_{2}=[\frac{I_{2}}{2}(2+\theta_{1}^{*}-\theta_{2})]$ and
$\displaystyle\theta_{1}^{*}=\frac{2}{\mathcal{L}_{*}}\int_{0}^{t_{1}^{*}}\frac{ds}{l(s)^{2}}$,
and $(\bar{t}_{1},\bar{v}_{1})=F(\tilde{t}_{1},\tilde{v}_{1})\in R_{1}^{+}$.
We introduce a new pair of variables $(\tau,\mathcal{I})$ defined on the upper
singular strips

$\tau=\begin{cases}I(\theta-\theta_{1}^{*})&\hbox{on $R_{1}^{+}$}\\\
I(\theta-\theta_{2}^{*})&\hbox{on $R_{2}^{+}$}\end{cases},\
\mathcal{I}=\frac{I}{\mathcal{L}_{*}}\hbox{ on $R_{1}^{+}$, $R_{2}^{+}$}$

Now we present the Poincaré maps $P^{12}_{UU}:R_{1}^{+}\to R_{2}^{+}$ and
$P^{21}_{UU}:R_{2}^{+}\to R_{1}^{+}$ which captures the collision dynamics
when the ball travels from one singular strips to the other. We need the
following constants $(i=1,2)$:

$\displaystyle\Delta_{i}=\frac{1}{2}\frac{l(t_{i}^{*}+)}{l(t_{i}^{*}-)}\left(l(t_{i}^{*}-)\dot{l}(t_{i}^{*}+)-l(t_{i}^{*}+)\dot{l}(t_{i}^{*}-)\right),$
$\displaystyle\Delta^{\prime}_{i}=\frac{1}{8}l(t_{i}^{*}+)^{2}\left(l(t_{i}^{*}-)\ddot{l}(t_{i}^{*}+)-l(t_{i}^{*}+)\ddot{l}(t_{i}^{*}-)\right),$
$\displaystyle\Delta^{\prime\prime}_{i}=\frac{1}{24}l(t_{i}^{*}-)l(t_{i}^{*}+)\left(l(t_{i}^{*}-)\ddot{l}(t_{i}^{*}+)-l(t_{i}^{*}+)\ddot{l}(t_{i}^{*}-)\right).$

######  Proposition 4.3 (Upper-Upper).

Suppose that $(\tau_{1},\mathcal{I}_{1})\in R_{1}^{+}$ and
$\mathcal{I}_{1}>V_{*}$, and that

$f(t^{*}_{2}+)-f(t^{*}_{2}-)\lesssim
l_{2}^{-}\\{\mathcal{L}_{*}\mathcal{I}_{1}(\theta_{2}^{*}-\theta_{1}^{*})-\tau_{1}\\}_{2}\lesssim
2-f(t^{*}_{2}+)-f(t^{*}_{2}-),$

where $\lesssim$ means the inequality holds up to an error of order
$\mathcal{O}(\frac{1}{\mathcal{I}})$, and $\\{\bullet\\}_{2}=\bullet\mod 2$.
Then the Poincaré map $P_{UU}^{12}:R_{1}^{+}\to R_{2}^{+}$ is given by
$(\bar{\tau}_{2},\bar{\mathcal{I}}_{2})=G_{UU}^{12}(\tau_{1},\mathcal{I}_{1})+H_{UU}^{12}(\tau_{1},\mathcal{I}_{1})+\mathcal{O}(\mathcal{I}_{1}^{-2})$
where

$G_{UU}^{12}(\tau_{1},\mathcal{I}_{1})=\left(-\frac{l_{2}^{-}}{l_{2}^{+}}\\{\mathcal{L}_{*}\mathcal{I}_{1}(\theta_{2}^{*}-\theta_{1}^{*})-\tau_{1}\\}_{2}+1+\frac{l_{2}^{-}}{l_{2}^{+}},\frac{l_{2}^{+}}{l_{2}^{-}}\mathcal{I}_{1}+\Delta_{2}(\bar{\tau}_{2}-1)\right)$

and

$H_{UU}^{12}(\tau_{1},\mathcal{I}_{1})=\left(0,\Delta^{\prime}_{2}(\bar{\tau}_{2}-1)^{2}/\mathcal{I}_{1}+\Delta^{\prime\prime}_{2}/\mathcal{I}_{1}\right)$

Similarly, suppose that $(\tau_{2},\mathcal{I}_{2})\in R_{2}^{+}$,
$\mathcal{I}_{2}>V_{*}$, and that

$f(t^{*}_{1}+)-f(t^{*}_{1}-)\lesssim
l_{1}^{-}\\{\mathcal{L}_{*}\mathcal{I}_{2}(2+\theta_{1}^{*}-\theta_{2}^{*})-\tau_{2}\\}_{2}\lesssim
2-f(t^{*}_{1}+)-f(t^{*}_{1}-).$

Then the Poincaré map $P_{UU}^{21}:R_{2}^{+}\to R_{1}^{+}$ is given by

$(\bar{\tau}_{1},\bar{\mathcal{I}}_{1})=G_{UU}^{21}(\tau_{2},\mathcal{I}_{2})+H_{UU}^{21}(\tau_{2},\mathcal{I}_{2})+\mathcal{O}(\mathcal{I}_{2}^{-2})$

where

$G_{UU}^{21}(\tau_{2},\mathcal{I}_{2})=\left(-\frac{l_{1}^{-}}{l_{1}^{+}}\\{\mathcal{L}_{*}\mathcal{I}_{2}(2+\theta_{1}^{*}-\theta_{2}^{*})-\tau_{2}\\}_{2}+1+\frac{l_{1}^{-}}{l_{1}^{+}},\frac{l_{1}^{+}}{l_{1}^{-}}\mathcal{I}_{2}+\Delta_{1}(\bar{\tau}_{1}-1)\right),$
$H_{UU}^{21}(\tau_{2},\mathcal{I}_{2})=\left(0,\Delta^{\prime}_{1}(\bar{\tau}_{1}-1)^{2}/\mathcal{I}_{2}+\Delta^{\prime\prime}_{1}/\mathcal{I}_{2}\right)$

and $l_{i}^{\pm}=l(t_{i}^{*}\pm)$.

###### Proof.

We only derive the formula for $P_{UU}^{12}.$ The formula for $P_{UU}^{21}$
can be obtained in a similar fashion.

For the ease of notation we drop the sub/superscripts whenever they are clear
from the context. Note that near the jump discontinuity at $t^{*}$

$\displaystyle
l(\tilde{t}^{*})=l_{-}(t^{*})+\dot{l}_{-}(\tilde{t}-t^{*})+\frac{1}{2}\ddot{l}_{-}(t^{*})(\tilde{t}-t^{*})^{2}+\mathcal{O}(\tilde{v}^{-3}),$
$\displaystyle
l(\tilde{t}^{*})=l_{+}(t^{*})+\dot{l}_{+}(\bar{t}-t^{*})+\frac{1}{2}\ddot{l}_{+}(t^{*})(\bar{t}-t^{*})^{2}+\mathcal{O}(\tilde{v}^{-3}),$
$\displaystyle\dot{l}(\tilde{t}^{*})=\dot{l}_{-}(t^{*})+\ddot{l}_{-}(\tilde{t}-t^{*})+\mathcal{O}(\tilde{v}^{-2}),$
$\displaystyle\dot{l}(\tilde{t}^{*})=\dot{l}_{+}(t^{*})+\ddot{l}_{+}(\bar{t}-t^{*})+\mathcal{O}(\tilde{v}^{-2}),$
$\displaystyle\ddot{l}(\tilde{t}^{*})=\ddot{l}_{-}(t^{*})+\mathcal{O}(\tilde{v}^{-1}),$
$\displaystyle\ddot{l}(\tilde{t}^{*})=\ddot{l}_{+}(t^{*})+\mathcal{O}(\tilde{v}^{-1}),$

and that

$\bar{v}=\tilde{v}-2\dot{l}(\bar{t}),\quad\bar{t}-\tilde{t}=\frac{l(\bar{t})+l(\tilde{t})}{\bar{v}}.$

Hence by solving iteratively the implicit equation we attain

$\bar{t}-\tilde{t}=\frac{l_{+}+l_{-}}{\tilde{v}}+(\dot{l}_{+}+\dot{l}_{-})\frac{\bar{t}-t^{*}}{\tilde{v}}-\frac{\dot{l}_{-}(l_{+}+l_{-})}{\tilde{v}^{2}}+\mathcal{O}(\tilde{v}^{-3}).$

By a straightforward but tedious computation we arrive at

$\displaystyle
2\mathcal{L}_{*}^{-1}\left(\frac{l_{-}}{l_{+}}\bar{I}-\tilde{I}\right)$
$\displaystyle=(l_{+}\dot{l}_{-}-l_{-}\dot{l}_{+})-\frac{l_{+}\dot{l}_{-}-l_{-}\dot{l}_{+}}{l_{+}}(\bar{t}-t^{*})\bar{v}$
$\displaystyle\ \ \ \
+(l_{+}\ddot{l}_{-}-l_{-}\ddot{l}_{+}+\frac{l_{-}}{l_{+}}\dot{l}_{+}^{2}-\dot{l}_{-}\dot{l}_{+})(\bar{t}-t^{*})$
$\displaystyle\ \ \ \
+\left(\frac{l_{-}}{3}(l_{+}\ddot{l}_{+}-l_{-}\ddot{l}_{-})+\frac{\ddot{l}_{-}}{2}(l_{-}^{2}-l_{+}^{2})\right)\frac{1}{\bar{v}}$
$\displaystyle\ \ \ \
-\frac{l_{+}\ddot{l}_{-}-l_{-}\ddot{l}_{+}}{2l_{+}}(\bar{t}-t^{*})^{2}\bar{v}+\mathcal{O}(\tilde{v}^{-2})$
$\displaystyle=\frac{l_{-}\dot{l}_{+}-l_{+}\dot{l}_{-}}{l_{+}}\left((\bar{t}-t^{*})\bar{v}\big{(}1+\frac{\dot{l}_{+}}{\bar{v}}\big{)}-l_{+}\right)$
$\displaystyle\ \ \ \
+\frac{l_{-}\ddot{l}_{+}-l_{+}\ddot{l}_{-}}{2l_{+}\bar{v}}\left(\big{(}(\bar{t}-t^{*})\bar{v}-l_{+}\big{)}^{2}+\frac{l_{-}l_{+}(l^{-}\ddot{l}_{-}-l_{+}\ddot{l}_{+})}{3(l_{-}\ddot{l}_{+}-l_{+}\ddot{l}_{-})}\right)+\mathcal{O}(\tilde{v}^{-2}).$

It can be checked directly by Taylor expanding $\bar{I}$ and $\bar{\theta}$
that

$\bar{\tau}=\bar{I}(\bar{\theta}-\theta_{2}^{*})=\frac{1}{l_{+}}\left((\bar{t}-t^{*})\bar{v}+\dot{l}_{+}(\bar{t}-t^{*})\right)+\mathcal{O}(\tilde{v}^{-2}).$

Thus eventually we have

$\bar{\mathcal{I}}=(l_{+}/l_{-})\tilde{\mathcal{I}}+\Delta(\bar{\tau}-1)+\Delta^{\prime}(\bar{\tau}-1)^{2}/\tilde{\mathcal{I}}+\Delta^{\prime\prime}/\tilde{\mathcal{I}}+\mathcal{O}(\tilde{\mathcal{I}}^{-2}).$

Now we compute $\bar{\tau}$. Observe that

$\bar{\tau}=\frac{\bar{v}+\dot{l}_{+}}{l_{+}}(\bar{t}-t^{*})+\mathcal{O}(\bar{v}^{-2}),\
\tilde{I}(\tilde{\theta}-\tilde{\theta_{2}^{*}})=\frac{\tilde{v}+\dot{l}_{-}}{l_{-}}(\tilde{t}-t^{*})+\mathcal{O}(\bar{v}^{-2}).$

Therefore

$\displaystyle l_{+}\bar{\tau}$
$\displaystyle=((\bar{t}-\tilde{t})+(\tilde{t}-t^{*}))(\tilde{v}+\dot{l}_{-}-(\dot{l}_{-}+\dot{l}_{+}))+\mathcal{O}(\tilde{v}^{-2})$
$\displaystyle=(\tilde{t}-t^{*})(\tilde{v}+\dot{l}_{-})+(\bar{t}-\tilde{t})(\tilde{v}+\dot{l}_{-})-(\bar{t}-t^{*})(\dot{l}_{-}+\dot{l}_{+})+\mathcal{O}(\tilde{v}^{-2})$
$\displaystyle=l_{-}\tilde{I}(\tilde{\theta}-\tilde{\theta_{2}^{*}})+\frac{l_{-}+l_{+}}{\tilde{v}}(\tilde{v}+\dot{l}_{-})+(\dot{l}_{-}+\dot{l}_{+})(\bar{t}-t^{*})$
$\displaystyle-\frac{\dot{l}_{-}(l_{-}+l_{+})}{\tilde{v}}-(\bar{t}-t^{*})(\dot{l}_{-}+\dot{l}_{+})+\mathcal{O}(\tilde{v}^{-2})$
$\displaystyle=l_{-}\tilde{I}(\tilde{\theta}-\tilde{\theta_{2}^{*}})+l_{-}+l_{+}+\mathcal{O}(\tilde{v}^{-2}),$

which gives

$\bar{\tau}=(l_{-}/l_{+})\tilde{I}(\tilde{\theta}-\theta_{2}^{*})+1+l_{-}/l_{+}+\mathcal{O}(\tilde{I}^{-2}).$

But Lemma 4.1 implies that

$\tilde{I}=I+\mathcal{O}(I^{-2}),\ \
\tilde{\theta}=\theta+\frac{2n_{1}}{I}+\mathcal{O}(I^{-3})$

hence we have

$\tilde{I}(\tilde{\theta}-\theta_{2}^{*})=\tau+2n_{1}+I(\theta_{1}^{*}-\theta_{2}^{*})+\mathcal{O}(I^{-2})$

where $n_{1}=[\frac{I_{1}}{2}(\theta_{2}^{*}-\theta_{1})]$.
We hitherto complete the proof of the formula for $P_{UU}^{12}$. ∎

####  4.2.2. The Lower-Lower Chamber Case

We present here the mirror case to Section 4.2.1, i.e. when the ball stays in
the lower chamber both before and after it makes a jump.
We need the following constants $(i=1,2)$

$\displaystyle\zeta_{i}^{*}$
$\displaystyle=\frac{2}{M_{*}}\int_{0}^{t_{i}^{*}}m(s)^{-2}ds$
$\displaystyle\Upsilon_{i}$
$\displaystyle=\frac{1}{2}\frac{m_{i}^{+}}{m_{i}^{-}}(m_{i}^{-}\dot{m}_{i}^{+}-m_{i}^{+}\dot{m}_{i}^{-})$
$\displaystyle\Upsilon_{i}^{\prime}$
$\displaystyle=\frac{1}{8}m_{i}^{+2}(m_{i}^{-}\ddot{m}_{i}^{+}-m_{i}^{+}\ddot{m}_{i}^{-})$
$\displaystyle\Upsilon_{i}^{\prime\prime}$
$\displaystyle=\frac{1}{24}m_{i}^{-}m_{i}^{+}(m_{i}^{-}\ddot{m}_{i}^{-}-m_{i}^{+}\ddot{m}_{i}^{+})$

We introduce a new pair of variables $(\rho,\mathcal{J})$ on the lower
singular strips, which is the counterpart of $(\tau,\mathcal{I})$ as follows

$\rho=\begin{cases}J(\zeta-\zeta_{1}^{*})&\hbox{on $R_{1}^{-}$}\\\
J(\zeta-\zeta_{2}^{*})&\hbox{on $R_{2}^{-}$}\end{cases},\
\mathcal{J}=\frac{J}{\mathcal{M}_{*}}\hbox{ on $R_{1}^{-}$,$R_{2}^{-}$}$

######  Proposition 4.4 (Lower-Lower).

Suppose that $(\rho_{1},\mathcal{J}_{1})\in R_{1}^{-}$, and
$\mathcal{J}_{1}>V_{*}$, and that

$f(t^{*}_{2}-)-f(t^{*}_{2}+)\lesssim-
m_{2}^{-}\\{\mathcal{M}_{*}\mathcal{J}_{1}(\zeta_{2}^{*}-\zeta_{1}^{*})-\rho_{1}\\}_{2}\lesssim
f(t^{*}_{2}-)+f(t^{*}_{2}+).$

Then the Poincaré map $P_{LL}^{12}:R_{1}^{-}\to R_{2}^{-}$ is given by

$(\bar{\rho}_{2},\bar{\mathcal{J}}_{2})=G_{LL}^{12}(\rho_{1},\mathcal{J}_{1})+H_{LL}^{12}(\rho_{1},\mathcal{J}_{1})++\mathcal{O}(\mathcal{J}_{1}^{-2})$

where

$G_{LL}^{12}(\rho_{1},\mathcal{J}_{1})=\left(-\frac{m_{2}^{-}}{m_{2}^{+}}\\{\mathcal{M}_{*}\mathcal{J}_{1}(\zeta_{2}^{*}-\zeta_{1}^{*})-\rho_{1}\\}_{2}+1+\frac{m_{2}^{-}}{m_{2}^{+}},\frac{m_{2}^{+}}{m_{2}^{-}}\mathcal{J}_{1}+\Upsilon_{2}(\bar{\rho}_{2}-1)\right)$

and

$H_{LL}^{12}(\rho_{1},\mathcal{J}_{1})=\left(0,\Upsilon^{\prime}_{2}(\bar{\rho}_{2}-1)^{2}/\mathcal{J}_{1}+\Upsilon^{\prime\prime}_{2}/\mathcal{J}_{1}\right).$

Similarly, suppose that $(\rho_{2},\mathcal{J}_{2})\in R_{2}^{-}$ and
$\mathcal{J}_{2}>V_{*}$, and that

$f(t^{*}_{1}-)-f(t^{*}_{1}+)\lesssim-
m_{1}^{-}\\{\mathcal{M}_{*}\mathcal{J}_{2}(2+\zeta_{1}^{*}-\zeta_{2}^{*})-\rho_{2}\\}_{2}\lesssim
f(t^{*}_{1}-)+f(t^{*}_{1}+).$

Then the Poincaré map $P_{LL}^{21}:R_{2}^{-}\to R_{1}^{-}$ is given by

$(\bar{\rho}_{1},\bar{\mathcal{J}}_{1})=G_{LL}^{21}(\rho_{2},\mathcal{J}_{2})+H_{LL}^{21}(\rho_{2},\mathcal{J}_{2})++\mathcal{O}(\mathcal{J}_{2}^{-2})$

where

$G_{LL}^{21}(\rho_{2},\mathcal{J}_{2})=\left(-\frac{m_{1}^{-}}{m_{1}^{+}}\\{\mathcal{M}_{*}\mathcal{J}_{2}(2+\zeta_{1}^{*}-\zeta_{2}^{*})-\rho_{2}\\}_{2}+1+\frac{m_{1}^{-}}{m_{1}^{+}},\frac{m_{1}^{+}}{m_{1}^{-}}\mathcal{J}_{2}+\Upsilon_{1}(\bar{\rho}_{1}-1)\right),$

$H_{LL}^{21}(\rho_{2},\mathcal{J}_{2})=\left(0,\Upsilon^{\prime}_{1}(\bar{\rho}_{1}-1)^{2}/\mathcal{J}_{2}+\Upsilon^{\prime\prime}_{1}/\mathcal{J}_{2}\right),$

and $m_{i}^{\pm}=m(t_{i}^{*}\pm)$.

####  4.2.3. The Upper-Lower Chamber Case

Now we suppose that $(\tilde{t},\tilde{v})\in\tilde{R}$ and that the ball is
in the upper chamber. Also we assume that the next wall is above the previous
one when the ball passes through the singularity at $t=t_{*}$:
$f(t_{*}-)<f(t_{*}+)$. Let $(\bar{t},\bar{v})=F(\tilde{t},\tilde{v})$.

If $v(t_{*}-t)>2-f(t)-f(t_{*}+)$, then the ball collides with the ceiling and
then enters the lower chamber (c.f. Figure 4 on the left).
Rather than resorting to the detailed computation as we have done in the
constant chamber cases, we insert an imaginary stationary slit, whose length
is negligible, at the height
$f_{*}=1-\tilde{v}(t_{*}-\tilde{t})+l(\tilde{t})$, so that the two consecutive
collisions at the moving slits are concatenated by two fictional collisions at
the imaginary wall, to which the  (Upper-Upper). and  (Lower-Lower). formulas
readily apply.

$(\tilde{t},\tilde{v})$($\bar{t},\bar{v})$$(t_{*},v_{*})$ Figure 7. The
Imaginary Stationary Wall

More precisely, as the ball leaves the previous slit at time $\tilde{t}$ with
velocity $\tilde{v}$, it collides against the imaginary tiny slit at time
$t_{*}$ and the outgoing velocity is still $v_{*}=\tilde{v}$ as the slit is
stationary. Meanwhile we also imagine that the ball leaves from below the
fictional slit at time $t_{*}$ with velocity $v_{*}=-\tilde{v}$ (with an abuse
of notation), then it collides at the next moving slit at time $\bar{t}$ with
outgoing velocity $\bar{v}$.
Let us denote $I_{*}=I(t_{*},v_{*})$, etc. We will need the following
constants

$\displaystyle\kappa_{Ii}=\frac{1}{2}m_{+}(\dot{m}_{+}-m_{+}\dot{l}_{-}/l_{-})$
$\displaystyle\kappa_{Ii}^{\prime}=\frac{1}{2}m_{+}\dot{l}_{-}/l_{-}$
$\displaystyle\kappa_{Ii}^{\prime\prime}=\frac{1}{8}m_{+}\ddot{l}_{-}(1-\frac{1}{3}l_{-}^{2})$
$\displaystyle\kappa_{Ii}^{\prime\prime\prime}=\frac{1}{4}m_{+}^{2}(\ddot{l}_{-}+\frac{1}{6}l_{-}\ddot{m}_{+})$
$\displaystyle\kappa_{Ii}^{\prime\prime\prime\prime}=\frac{1}{8}m_{+}^{3}\ddot{l}_{-}$
$\displaystyle\kappa_{Ii}^{\prime\prime\prime\prime\prime}=\frac{1}{8}m_{+}^{2}\ddot{m}_{+}l_{-}$

where $i$ indicates that $l(t)$ and $m(t)$ are evaluated at $t=t_{i}^{*}$
($i=1,2$).
Then the dynamics between the singular strips is captured by the following
formula:

######  Proposition 4.5 (Upper-Lower I).

Assume that $(\tau_{1},\mathcal{I}_{1})\in R_{1}^{+}$ with
$\mathcal{I}_{1}>V_{*}$ and that

$l_{2}^{-}\\{\mathcal{L}_{*}\mathcal{I}_{1}(\theta_{2}^{*}-\theta_{1}^{*})-\tau_{1}\\}_{2}\gtrsim
2-f(t^{*}_{2}+)-f(t^{*}_{2}-).$

Then the Poincaré map $P_{UL\rm I}^{12}:R_{1}^{+}\to R_{2}^{-}$ is given by

$(\bar{\rho}_{2},\bar{\mathcal{J}}_{2})=G_{UL\rm
I}^{12}(\tau_{1},\mathcal{I}_{1})+H_{UL\rm
I}^{12}(\tau_{1},\mathcal{I}_{1})+\mathcal{O}(\mathcal{I}_{1}^{-2})$

where

$\displaystyle G_{UL\rm I}^{12}(\tau_{1},\mathcal{I}_{1})=\bigg{(}$
$\displaystyle\frac{l_{2}^{-}}{m_{2}^{+}}\\{\mathcal{L}_{*}\mathcal{I}_{1}(\theta_{2}^{*}-\theta_{1}^{*})-\tau_{1}\\}_{2}+\frac{m_{2}^{+}-l_{2}^{-}-1}{m_{2}^{+}},$
$\displaystyle-\frac{m_{2}^{+}}{l_{2}^{-}}\mathcal{I}_{1}+\kappa_{I2}(\bar{\rho}_{2}-1)-\kappa_{I2}^{\prime}\bigg{)}$

and

$H_{UL\rm
I}^{12}(\tau_{1},\mathcal{I}_{1})=\left(0,\frac{\kappa_{I2}^{\prime\prime}}{\mathcal{I}_{1}}+\kappa_{I2}^{\prime\prime\prime}\frac{\bar{\rho}_{2}-1}{I_{1}}+\kappa_{I2}^{\prime\prime\prime\prime}\frac{(\bar{\rho}_{2}-1)^{2}}{\mathcal{I}_{1}}-\kappa_{I2}^{\prime\prime\prime\prime\prime}\frac{(\bar{\rho}_{2}-1)^{3}}{\mathcal{I}_{1}}\right).$

Similarly, assume that $(\tau_{2},\mathcal{I}_{2})\in R_{2}^{+}$ with
$\mathcal{I}_{2}>V_{*}$ and that

$l_{1}^{-}\\{\mathcal{L}_{*}\mathcal{I}_{2}(2+\theta_{1}^{*}-\theta_{2}^{*})-\tau_{2}\\}_{2}\gtrsim
2-f(t^{*}_{1}+)-f(t^{*}_{1}-).$

Then the Poincaré map $P_{UL\rm I}^{21}:R_{2}^{+}\to R_{1}^{-}$ is given by

$(\bar{\rho}_{1},\bar{\mathcal{J}}_{1})=G_{UL\rm
I}^{21}(\tau_{2},\mathcal{I}_{2})+H_{UL\rm
I}^{21}(\tau_{2},\mathcal{I}_{2})+\mathcal{O}(\mathcal{I}_{2}^{-2})$

where

$\displaystyle G_{UL\rm I}^{21}(\tau_{2},\mathcal{I}_{2})=\bigg{(}$
$\displaystyle\frac{l_{1}^{-}}{m_{1}^{+}}\\{\mathcal{L}_{*}\mathcal{I}_{2}(2+\theta_{1}^{*}-\theta_{2}^{*})-\tau_{2}\\}_{2}+\frac{m_{1}^{+}-l_{1}^{-}-1}{m_{1}^{+}},$
$\displaystyle-\frac{m_{1}^{+}}{l_{1}^{-}}\mathcal{I}_{2}+\kappa_{I1}(\bar{\rho}_{1}-1)-\kappa_{I1}^{\prime}\bigg{)}$

and

$H_{UL\rm
I}^{21}(\tau_{2},\mathcal{I}_{2})=\left(0,\frac{\kappa_{I1}^{\prime\prime}}{\mathcal{I}_{2}}+\kappa_{I1}^{\prime\prime\prime}\frac{\bar{\rho}_{1}-1}{\mathcal{I}_{2}}+\kappa_{I1}^{\prime\prime\prime\prime}\frac{(\bar{\rho}_{1}-1)^{2}}{\mathcal{I}_{2}}-\kappa_{I1}^{\prime\prime\prime\prime\prime}\frac{(\bar{\rho}_{1}-1)^{3}}{\mathcal{I}_{2}}\right).$

###### Proof.

We present the proof of $P_{UL\rm I}^{12}.$ The formula for $P_{UL\rm I}^{21}$
can be obtained similarly. We suppress the sub/superscripts whenever they are
clear from the context.
We imagine that the ball collides at the fictional stationary wall at time
$t_{*}$ with outgoing velocity $v_{*}=\tilde{v}$. Then
$l_{*}=1-f_{*}=\tilde{v}(t_{*}-\tilde{t})-l(\tilde{t})$ and we have

$I_{*}=\frac{L_{*}}{2}l_{*}\tilde{v},\ \tau_{*}=0.$

From the  (Upper-Upper). formula,

$\tau_{*}=\frac{l_{-}}{l_{*}}\tilde{I}(\tilde{\theta}-\theta_{2}^{*})+1+\frac{l_{-}}{l_{*}}+\mathcal{O}(v^{-2})$
$I_{*}=\frac{l_{*}}{l_{-}}\tilde{I}+\frac{L_{*}l_{*}^{2}}{2l_{-}}\dot{l}_{-}-\frac{L_{*}^{2}l_{*}^{3}\ddot{l}_{-}}{8\tilde{I}}+\frac{L_{*}^{2}l_{*}^{2}l_{*}\ddot{l}_{-}}{24\tilde{I}}+\mathcal{O}(v^{-2}).$

Now we imagine that the ball leaves from below the fictional stationary wall
at time $t_{*}$ with outgoing velocity $v_{*}=-\tilde{v}$ (with an abuse of
notation). Then $m_{*}=-m(\bar{t})-v(\bar{t}-t_{*})$ and we have

$J_{*}=-\frac{M_{*}}{2}m_{*}\tilde{v},\ \rho_{*}=0.$

From the  (Lower-Lower). formula,

$\bar{\rho}=1+\frac{m_{*}}{m_{+}}+\mathcal{O}(v^{-2})$
$\bar{J}=\frac{m_{+}}{m_{*}}J_{*}+\frac{M_{*}}{2}m_{+}\dot{m}_{+}(\bar{\rho}-1)+\frac{M_{*}^{2}}{8J_{*}}m_{+}^{2}m_{*}\ddot{m}_{+}(\bar{\rho}-1)^{2}-\frac{M_{*}^{2}}{24J_{*}}m_{+}^{2}m_{*}\ddot{m}_{+}+\mathcal{O}(v^{-2}).$

We observe that

$\displaystyle l_{*}$ $\displaystyle=v(t_{*}-\tilde{t})-l(\tilde{t})$
$\displaystyle=v(t_{*}-\tilde{t})-l_{-}-\dot{l}_{-}(\tilde{t}-t_{*})+\mathcal{O}(v^{-2})$
$\displaystyle=-(\tilde{v}+\dot{l}_{-})(t_{*}-\tilde{t})-l_{-}+\mathcal{O}(v^{-2})$
$\displaystyle=-l_{-}\tilde{I}(\tilde{\theta}-\theta_{2}^{*})-l_{-}+\mathcal{O}(v^{-2})$

and that

$\displaystyle m_{*}$ $\displaystyle=-m(\bar{t})-v(\bar{t}-t_{*})$
$\displaystyle=-m_{+}-\dot{m}_{+}(\bar{t}-t_{*})+(\bar{v}+2\dot{m}_{+})(\bar{t}-t_{*})+\mathcal{O}(v^{-2})$
$\displaystyle=-m_{+}+(\bar{v}+\dot{m}_{+})(\bar{t}-t_{*})+\mathcal{O}(v^{-2})$
$\displaystyle=-m_{+}+m_{+}\bar{\rho}+\mathcal{O}(v^{-2})$

Since $m_{*}=l_{*}-1$,

$\bar{\rho}=-\frac{l_{-}}{m_{+}}\tilde{I}(\tilde{\theta}-\theta_{2}^{*})+\frac{m_{+}-l_{-}-1}{m_{+}}+\mathcal{O}(v^{-2}).$

Finally the relation between $I_{*}$ and $J_{*}$, together with Lemma 4.1,
produces the formula for $\bar{J}$. ∎

If $v(t_{*}-t)<f(t_{*}+)-f(t)$, then the ball enters the lower chamber
immediately after it leaves the previous slit (c.f. Figure 3 on the right).
The imaginary wall trick no longer applies, so we have to return to the direct
computation.
We will need the following constants

$\displaystyle\kappa_{\rm
IIi}^{\prime\prime}=\frac{1}{4}m_{+}^{2}\ddot{l}_{-}$
$\displaystyle\kappa_{\rm
IIi}^{\prime\prime\prime}=\frac{1}{8}m_{+}^{2}(l_{-}\ddot{m}_{+}-m_{+}\ddot{l}_{-})$
$\displaystyle\kappa_{\rm
IIi}^{\prime\prime\prime\prime}=\frac{1}{24}m_{+}l_{-}(l_{-}^{2}\ddot{l}_{-}-l_{-}m_{+}\ddot{m}_{+}-3\ddot{l}_{-})$

where $i$ indicates that $l(t)$ and $m(t)$ are evaluated at $t=t_{i}^{*}$
($i=1,2$).

######  Proposition 4.6 (Upper-Lower II).

Assume that $(\tau_{1},\mathcal{I}_{1})\in R_{1}^{+}$ with
$\mathcal{I}_{1}>V_{*}$ and that

$f(t^{*}_{2}+)-f(t^{*}_{2}-)\gtrsim
l_{2}^{-}\\{\mathcal{L}_{*}\mathcal{I}_{1}(\theta_{2}^{*}-\theta_{1}^{*})-\tau_{1}\\}_{2}.$

Then the Poincaré map $P_{UL\rm II}^{12}:R_{1}^{+}\to R_{2}^{-}$ is given by

$(\bar{\rho}_{2},\bar{\mathcal{J}}_{2})=G_{UL\rm
II}^{12}(\tau_{1},\mathcal{I}_{1})+H_{UL\rm
II}^{12}(\tau_{1},\mathcal{I}_{1})+\mathcal{O}(\mathcal{I}_{1}^{-2})$

where

$\displaystyle G_{UL\rm II}^{12}(\tau_{1},\mathcal{I}_{1})=\bigg{(}$
$\displaystyle\frac{l_{2}^{-}}{m_{2}^{+}}\\{\mathcal{L}_{*}\mathcal{I}_{1}(\theta_{2}^{*}-\theta_{1}^{*})-\tau_{1}\\}_{2}+\frac{m_{2}^{+}-l_{2}^{-}+1}{m_{2}^{+}},$
$\displaystyle-\frac{m_{2}^{+}}{l_{2}^{-}}\mathcal{\mathcal{I}}_{1}+\kappa_{\rm
I2}(\bar{\rho}_{2}-1)+\kappa_{\rm I2}^{\prime}\bigg{)}$

and

$H_{UL\rm II}^{12}(\tau_{1},\mathcal{I}_{1})=\left(0,-\kappa_{\rm
II2}^{\prime\prime}\frac{\bar{\rho}_{2}-1}{\mathcal{I}_{1}}-\kappa_{\rm
II2}^{\prime\prime\prime}\frac{(\bar{\rho}_{2}-1)^{2}}{\mathcal{I}_{1}}-\frac{\kappa_{\rm
II2}^{\prime\prime\prime\prime}}{\mathcal{I}_{1}}\right).$

Similarly, assume that $(\tau_{2},\mathcal{I}_{2})\in R_{2}^{+}$ with
$\mathcal{I}_{2}>V_{*}$ and that

$f(t^{*}_{1}+)-f(t^{*}_{1}-)\gtrsim
l_{1}^{-}\\{\mathcal{L}_{*}\mathcal{I}_{2}(2+\theta_{1}^{*}-\theta_{2}^{*})-\tau_{2}\\}_{2}.$

Then the Poincaré map $P_{UL\rm II}^{21}:R_{2}^{+}\to R_{1}^{-}$ is given by

$(\bar{\rho}_{1},\bar{\mathcal{J}}_{1})=G_{UL\rm
II}^{21}(\tau_{2},\mathcal{I}_{2})+H_{UL\rm
II}^{21}(\tau_{2},\mathcal{I}_{2})+\mathcal{O}(\mathcal{I}_{2}^{-2})$

where

$\displaystyle G_{UL\rm II}^{21}(\tau_{2},\mathcal{I}_{2})=\bigg{(}$
$\displaystyle\frac{l_{1}^{-}}{m_{1}^{+}}\\{\mathcal{L}_{*}\mathcal{I}_{2}(2+\theta_{1}^{*}-\theta_{2}^{*})-\tau_{2}\\}_{2}+\frac{m_{1}^{+}-l_{1}^{-}+1}{m_{1}^{+}},$
$\displaystyle-\frac{m_{1}^{+}}{l_{1}^{-}}\mathcal{I}_{2}+\kappa_{\rm
I1}(\bar{\rho}_{1}-1)+\kappa_{\rm I1}^{\prime}\bigg{)}$

and

$H_{UL\rm II}^{21}(\tau_{2},\mathcal{I}_{2})=\left(0,-\kappa_{\rm
II1}^{\prime\prime}\frac{\bar{\rho}_{1}-1}{\mathcal{I}_{2}}-\kappa_{\rm
II1}^{\prime\prime\prime}\frac{(\bar{\rho}_{1}-1)^{2}}{\mathcal{I}_{2}}-\frac{\kappa_{\rm
II1}^{\prime\prime\prime\prime}}{\mathcal{I}_{2}}\right).$

###### Proof.

Again we only prove the formula for $P_{UL\rm II}^{12}$.
We have from Equation 5 that

$\displaystyle v(\bar{t}-\tilde{t})$ $\displaystyle=f(\bar{t})-f(\tilde{t})$
$\displaystyle\implies\tilde{v}((\bar{t}-t_{*})-(\tilde{t}-t_{*}))$
$\displaystyle=-m(\bar{t})+l(\tilde{t})-1$ $\displaystyle\implies
m_{+}-(\bar{v}+\dot{m}_{+})(\bar{t}-t_{*})$
$\displaystyle=(\tilde{v}+\dot{l}_{-})(\tilde{t}-t_{*})+l_{-}-1+\mathcal{O}(v^{-2})$
$\displaystyle\implies m_{+}-m_{+}\bar{\rho}$
$\displaystyle=l_{-}\tilde{I}(\tilde{\theta}-\theta_{2}^{*})+l_{-}-1+\mathcal{O}(v^{-2})$
$\displaystyle\implies\bar{\rho}$
$\displaystyle=-\frac{l_{-}}{m_{+}}\tilde{I}(\tilde{\theta}-\theta_{2}^{*})+\frac{m_{+}-l_{-}+1}{m_{+}}+\mathcal{O}(v^{-2})$

The computation is similar to that in the proof of Proposition 4.3. So we just
list the key steps.
We observe that

$\displaystyle 2\mathcal{M}_{*}^{-1}\bar{J}$
$\displaystyle=\bar{m}\bar{v}+\bar{m}\dot{\bar{m}}+\frac{\bar{m}^{2}\ddot{\bar{m}}}{3\bar{v}}$
$\displaystyle=m_{+}\bar{v}+m_{+}\dot{m}_{+}+m_{+}\dot{m}_{+}\bar{\rho}+m_{+}\ddot{m}_{+}(\bar{t}-t_{*})+\frac{m_{+}^{2}\ddot{m}_{+}}{3\bar{v}}$
$\displaystyle\ \ \ \ \
+\frac{\ddot{m}_{+}}{2}(\bar{t}-t_{*})^{2}\bar{v}+\mathcal{O}(v^{-2})$

and that

$\displaystyle 2\mathcal{L}_{*}^{-1}\tilde{I}$
$\displaystyle=\tilde{l}\tilde{v}+\tilde{l}\dot{\tilde{l}}+\frac{\tilde{l}^{2}\ddot{\tilde{l}}}{3\tilde{v}}$
$\displaystyle=l_{-}\tilde{v}+l_{-}\dot{l}_{-}+l_{-}\dot{l}_{-}\tilde{I}(\tilde{\theta}-\theta_{2}^{*})+l_{-}\ddot{l}_{-}(\tilde{t}-t_{*})+\frac{l_{-}^{2}\ddot{l}_{-}}{3\tilde{v}}$
$\displaystyle\ \ \ \ \
+\frac{\ddot{l}_{-}}{2}(\tilde{t}-t_{*})^{2}\tilde{v}+\mathcal{O}(v^{-2}).$

We also note from Equation 5 that

$\tilde{v}=-\bar{v}-2\dot{m}_{+}-2\ddot{m}_{+}(\bar{t}-t_{*})+\mathcal{O}(v^{-2})$

and that

$\displaystyle\bar{t}-\tilde{t}$
$\displaystyle=\frac{f(\bar{t})-f(\tilde{t})}{\tilde{v}}=\frac{-m(\bar{t})+l(\tilde{t})-1}{\tilde{v}}$
$\displaystyle=\frac{-m_{+}+l_{-}-1}{\tilde{v}}+(-\dot{m}_{+}+\dot{l}_{-})\frac{\bar{t}-t_{*}}{\tilde{v}}-\dot{l}_{-}\frac{\bar{t}-\tilde{t}}{\tilde{v}}+\mathcal{O}(v^{-3})$
$\displaystyle=\frac{-m_{+}+l_{-}-1}{\tilde{v}}+(-\dot{m}_{+}+\dot{l}_{-})\frac{\bar{t}-t_{*}}{\tilde{v}}+\dot{l}_{-}\frac{m_{+}-l_{-}+1}{\tilde{v}^{2}}+\mathcal{O}(v^{-3}).$

Therefore

$\displaystyle 2\mathcal{L}_{*}^{-1}\tilde{I}$
$\displaystyle=-l_{-}\bar{v}-2l_{-}\dot{m}_{+}+m_{+}\dot{l}_{-}+m_{+}\dot{l}_{-}\bar{\rho}+\dot{l}_{-}$
$\displaystyle\ \ \ \ \
+(m_{+}\ddot{l}_{-}-2l_{-}\ddot{m}_{+}+\ddot{l}_{-})(\bar{t}-t_{*})-\frac{\ddot{l}_{-}}{2}(\bar{t}-t_{*})^{2}\bar{v}$
$\displaystyle\ \ \ \ \
-\frac{\ddot{l}_{-}}{2\bar{v}}\left((m_{+}+1)^{2}-\frac{l_{-}^{2}}{3}\right)+\mathcal{O}(v^{-2})$

hence

$\displaystyle\frac{2l_{-}}{\mathcal{M}_{*}}\bar{J}+\frac{2m_{+}}{\mathcal{L}_{*}}\tilde{I}$
$\displaystyle=m_{+}\dot{l}_{-}+m_{+}(l_{-}\dot{m}_{+}-m_{+}\dot{l}_{-})(\bar{\rho}-1)-\frac{1}{2}\mathcal{L}_{*}l_{-}m_{+}^{2}\ddot{l}_{-}\frac{\bar{\rho}-1}{I}$
$\displaystyle\ \ \ \ \
-\frac{1}{4}\mathcal{L}_{*}l_{-}m_{+}^{2}(l_{-}\ddot{m}_{+}-m_{+}\ddot{l}_{-})\frac{(\bar{\rho}-1)^{2}}{I}$
$\displaystyle\ \ \ \ \
-\frac{1}{12I}\mathcal{L}_{*}l_{-}m_{+}(l_{-}^{2}\ddot{l}_{-}-l_{-}m_{+}\ddot{m}_{+}-3\ddot{l}_{-})+\mathcal{O}(v^{-2})$

which produces the desired formula together with Lemma 4.1. ∎

####  4.2.4. The Lower-Upper Chamber Case

Finally we suppose that $(\tilde{t},\tilde{v})\in\tilde{R}$ and that the ball
is in the lower chamber. Also we assume that the next wall is below the
previous one when the ball passes through the singularity at $t=t_{*}$:
$f(t_{*}-)>f(t_{*}+)$. Again let $(\bar{t},\bar{v})=F(\tilde{t},\tilde{v})$.

If $-v(t_{*}-t)>f(t)+f(t_{*}-)$, then the ball collides with the floor and
enters the upper chamber (c.f. Figure 4 on the left).
The imaginary stationary wall trick also applies in this case, which produces
a desired formula with the following constants

$\displaystyle\chi_{Ii}=\frac{1}{2}l_{+}(\dot{l}_{+}-l_{+}\dot{m}_{-}/m_{-})$
$\displaystyle\chi_{Ii}^{\prime}=\frac{1}{2}l_{+}\dot{m}_{-}/m_{-}$
$\displaystyle\chi_{Ii}^{\prime\prime}=\frac{1}{8}l_{+}\ddot{m}_{-}(1-\frac{1}{3}m_{-}^{2})$
$\displaystyle\chi_{Ii}^{\prime\prime\prime}=\frac{1}{4}l_{+}^{2}(\frac{1}{6}m_{-}\ddot{l}_{+}-\ddot{m}_{-})$
$\displaystyle\chi_{Ii}^{\prime\prime\prime\prime}=\frac{1}{8}l_{+}^{3}\ddot{m}_{-}$
$\displaystyle\chi_{Ii}^{\prime\prime\prime\prime\prime}=\frac{1}{8}l_{+}^{2}\ddot{l}_{+}m_{-}$

where $i$ indicates that $l(t)$ and $m(t)$ are evaluated at $t=t_{i}^{*}$
($i=1,2$).

######  Proposition 4.7 (Lower-Upper I).

Assume that $(\rho_{1},\mathcal{J}_{1})\in R_{1}^{-}$ with
$\mathcal{J}_{1}>V_{*}$ and that
$-m_{2}^{-}\\{\mathcal{M}_{*}\mathcal{J}_{1}(\zeta_{2}^{*}-\zeta_{1}^{*})-\rho_{1}\\}_{2}\gtrsim
f(t^{*}_{2}-)+f(t^{*}_{2}+)$. Then the Poincaré map $P_{LU\rm
I}^{12}:R_{1}^{-}\to R_{2}^{+}$ is given by

$(\bar{\tau}_{2},\bar{\mathcal{I}}_{2})=G_{LU\rm
I}^{12}(\rho_{1},\mathcal{J}_{1})+H_{LU\rm
I}^{12}(\rho_{1},\mathcal{J}_{1})+\mathcal{O}(\mathcal{J}_{1}^{-2})$

where

$\displaystyle G_{LU\rm I}^{12}(\rho_{1},\mathcal{J}_{1})=\bigg{(}$
$\displaystyle\frac{m_{2}^{-}}{l_{2}^{+}}\\{\mathcal{M}_{*}\mathcal{J}_{1}(\zeta_{2}^{*}-\zeta_{1}^{*})-\rho_{1}\\}_{2}+\frac{l_{2}^{+}-m_{2}^{-}+1}{l_{2}^{+}},$
$\displaystyle-\frac{l_{2}^{+}}{m_{2}^{-}}\mathcal{J}_{1}+\chi_{I2}(\bar{\tau}_{2}-1)+\chi_{I2}^{\prime}\bigg{)}$

and

$H_{LU\rm
I}^{12}(\rho_{1},\mathcal{J}_{1})=\left(0,\frac{\chi_{I2}^{\prime\prime}}{\mathcal{J}_{1}}+\chi_{I2}^{\prime\prime\prime}\frac{\bar{\tau}_{2}-1}{\mathcal{J}_{1}}+\chi_{I2}^{\prime\prime\prime\prime}\frac{(\bar{\tau}_{2}-1)^{2}}{\mathcal{J}_{1}}-\chi_{I2}^{\prime\prime\prime\prime\prime}\frac{(\bar{\tau}_{2}-1)^{3}}{\mathcal{J}_{1}}\right).$

Similarly, assume that $(\rho_{2},\mathcal{J}_{2})\in R_{2}^{-}$ with
$\mathcal{J}_{2}>V_{*}$ and that

$-m_{1}^{-}\\{\mathcal{M}_{*}\mathcal{J}_{2}(2+\zeta_{1}^{*}-\zeta_{2}^{*})-\rho_{1}\\}_{2}\gtrsim
f(t^{*}_{1}-)+f(t^{*}_{1}+).$

Then the Poincaré map $P_{LU\rm I}^{21}:R_{2}^{-}\to R_{1}^{+}$ is given by

$(\bar{\tau}_{1},\bar{\mathcal{I}}_{1})=G_{LU\rm
I}^{21}(\rho_{2},\mathcal{J}_{2})+H_{LU\rm
I}^{21}(\rho_{2},\mathcal{J}_{2})+\mathcal{O}(\mathcal{J}_{2}^{-2})$

where

$\displaystyle G_{LU\rm I}^{21}(\rho_{2},\mathcal{J}_{2})=\bigg{(}$
$\displaystyle\frac{m_{1}^{-}}{l_{1}^{+}}\\{\mathcal{M}_{*}\mathcal{J}_{2}(2+\zeta_{1}^{*}-\zeta_{2}^{*})-\rho_{2}\\}_{2}+\frac{l_{1}^{+}-m_{1}^{-}+1}{l_{1}^{+}},$
$\displaystyle\frac{l_{1}^{+}}{m_{1}^{-}}\mathcal{J}_{2}+\chi_{I1}(\bar{\tau}_{1}-1)+\chi_{I1}^{\prime}\bigg{)}$

and

$H_{LU\rm
I}^{21}(\rho_{2},\mathcal{J}_{2})=\left(0,\frac{\chi_{I1}^{\prime\prime}}{\mathcal{J}_{2}}+\chi_{I1}^{\prime\prime\prime}\frac{\bar{\tau}_{1}-1}{\mathcal{J}_{2}}+\chi_{I1}^{\prime\prime\prime\prime}\frac{(\bar{\tau}_{1}-1)^{2}}{\mathcal{J}_{2}}-\chi_{I1}^{\prime\prime\prime\prime\prime}\frac{(\bar{\tau}_{1}-1)^{3}}{\mathcal{J}_{2}}\right).$

Next, if $-v(t_{*}-t)<f(t)-f(t_{*}-)$, then the ball enters the lower chamber
immediately after it leaves the previous slit (c.f. Figure 5 on the right).
The computation as we have performed for Proposition 4.6 can be reproduced
here to present the formula in this case, the proof of which we ergo omit. We
will need the following constants

$\displaystyle\chi_{\rm IIi}^{\prime\prime}=\frac{1}{4}l_{+}^{2}\ddot{m}_{-}$
$\displaystyle\chi_{\rm
IIi}^{\prime\prime\prime}=\frac{1}{8}l_{+}^{2}(m_{-}\ddot{l}_{+}-l_{+}\ddot{m}_{-})$
$\displaystyle\chi_{\rm
IIi}^{\prime\prime\prime\prime}=\frac{1}{24}l_{+}m_{-}(m_{-}^{2}\ddot{m}_{-}-m_{-}l_{+}\ddot{l}_{+}-3\ddot{m}_{-})$

where $i$ indicates that $l(t)$ and $m(t)$ are evaluated at $t=t_{i}^{*}$
($i=1,2$).

######  Proposition 4.8 (Lower-Upper II).

Assume that $(\rho_{1},\mathcal{J}_{1})\in R_{1}^{-}$ with
$\mathcal{J}_{1}>V_{*}$ and that $f(t^{*}_{2}-)-f(t^{*}_{2}+)\gtrsim-
m_{2}^{-}\\{\mathcal{M}_{*}\mathcal{J}_{1}(\zeta_{2}^{*}-\zeta_{1}^{*})-\rho_{1}\\}_{2}$.
Then the Poincaré map $P_{LU\rm II}^{12}:R_{1}^{-}\to R_{2}^{+}$ is given by

$(\bar{\tau}_{2},\bar{\mathcal{I}}_{2})=G_{LU\rm
II}^{12}(\rho_{1},\mathcal{J}_{1})+H_{LU\rm
II}^{12}(\rho_{1},\mathcal{J}_{1})+\mathcal{O}(\mathcal{J}_{1}^{-2})$

where

$\displaystyle G_{LU\rm II}^{12}(\rho_{1},\mathcal{J}_{1})=\bigg{(}$
$\displaystyle\frac{m_{2}^{-}}{l_{2}^{+}}\\{\mathcal{M}_{*}\mathcal{J}_{1}(\zeta_{2}^{*}-\zeta_{1}^{*})-\rho_{1}\\}_{2}+\frac{l_{2}^{+}-m_{2}^{-}-1}{l_{2}^{+}},$
$\displaystyle-\frac{l_{2}^{+}}{m_{2}^{-}}\mathcal{J}_{1}+\chi_{I2}(\bar{\tau}_{2}-1)-\chi_{I2}^{\prime}\bigg{)}$

and

$H_{LU\rm II}^{12}(\rho_{1},\mathcal{J}_{1})=\left(0,\chi_{\rm
II2}^{\prime\prime}\frac{\bar{\tau}_{2}-1}{\mathcal{J}_{1}}-\chi_{\rm
II2}^{\prime\prime\prime}\frac{(\bar{\tau}_{2}-1)^{2}}{\mathcal{J}_{1}}-\frac{\chi_{\rm
II2}^{\prime\prime\prime\prime}}{\mathcal{J}_{1}}\right).$

Similarly, assume that $(\rho_{2},\mathcal{J}_{2})\in R_{2}^{-}$ with
$\mathcal{J}_{2}>V_{*}$ and that

$f(t^{*}_{1}-)-f(t^{*}_{1}+)\gtrsim-
m_{1}^{-}\\{\mathcal{M}_{*}\mathcal{J}_{2}(2+\zeta_{1}^{*}-\zeta_{2}^{*})-\rho_{2}\\}_{2}.$

Then the Poincaré map $P_{LU\rm II}^{21}:R_{2}^{-}\to R_{1}^{+}$ is given by

$(\bar{\tau}_{1},\bar{\mathcal{I}}_{1})=G_{LU\rm
II}^{21}(\rho_{2},\mathcal{J}_{2})+H_{LU\rm
II}^{21}(\rho_{2},\mathcal{J}_{2})+\mathcal{O}(\mathcal{J}_{2}^{-2})$

where

$\displaystyle G_{LU\rm II}^{21}(\rho_{2},\mathcal{J}_{2})=\bigg{(}$
$\displaystyle\frac{m_{1}^{-}}{l_{1}^{+}}\\{\mathcal{J}_{2}(2+\zeta_{1}^{*}-\zeta_{2}^{*})-\rho_{2}\\}_{2}+\frac{l_{1}^{+}-m_{1}^{-}-1}{l_{1}^{+}},$
$\displaystyle-\frac{l_{1}^{+}}{m_{1}^{-}}\mathcal{J}_{2}+\chi_{\rm
I1}(\bar{\tau}_{1}-1)-\chi_{\rm I1}^{\prime}\bigg{)}$

and

$H_{LU\rm II}^{21}(\rho_{2},\mathcal{J}_{2})=\left(0,\chi_{\rm
II1}^{\prime\prime}\frac{\bar{\tau}_{1}-1}{\mathcal{J}_{2}}-\chi_{\rm
II1}^{\prime\prime\prime}\frac{(\bar{\tau}_{1}-1)^{2}}{\mathcal{J}_{2}}-\frac{\chi_{\rm
II1}^{\prime\prime\prime\prime}}{\mathcal{J}_{2}}\right).$

##  5\. Trapping Regions

In this section, we present the proof of Theorem 1. The assumptions in the
theorem lead to the creation of a trapping region where the ball gains energy
exponentially fast.

###### Proof.

We choose $V_{*}\gg 1$ so that the normal forms in Section 4 hold for
$|v|>V_{*}$. There are two cases.

(i) Suppose that $f_{L}(t_{1}^{*})<f_{R}(t_{1}^{*})$ and
$f_{L}(t_{2}^{*})>f_{R}(t_{2}^{*})$. The relative positions of the two slits
at two critical jumps trap the ball forever in the lower region once it
enters. Henceforth Proposition 4.4 predicts the change of energy after one
period in the lower chamber to be

$\bar{\bar{\mathcal{J}}}=\frac{m_{1}^{+}}{m_{1}^{-}}\frac{m_{2}^{+}}{m_{2}^{-}}\mathcal{J}+\mathcal{O}(1)$

Furthermore, the relative positions of the slits at two critical times
guarantee that $m(t_{1}^{*}+)<m(t_{1}^{*}-)<0$,
$m(t_{2}^{*}+)<m(t_{2}^{*}-)<0$, so the energy of the ball grows exponentially
fast at rate $\frac{m_{1}^{+}}{m_{1}^{-}}\frac{m_{2}^{+}}{m_{2}^{-}}>1$ in the
lower chamber.

$t_{1}^{*}$$t_{2}^{*}$ Figure 8. The Trapping Lower Chamber

If the ball starts from the lower chamber with $v_{0}<-V_{*}$, it enjoys
exponential energy growth with time immediately.
If the ball starts from the upper chamber with $v_{0}>V_{*}$, by Proposition
4.3 and the relative positions of the slits, its energy decreases at an
exponential rate $\frac{l_{1}^{+}}{l_{1}^{-}}\frac{l_{2}^{+}}{l_{2}^{-}}<1$
until it either enters the lower chamber or it enter the low energy region
$|v|<V_{*}$ and the normal form no longer applies. For any $V>V_{*}$, we
denote

$\mathcal{U}_{V}=\\{(t_{0},v_{0}):V<v_{0}<V+1,\ \limsup v_{n}<V_{*}\\}$

We claim that ${\mathrm{mes}}(\mathcal{U}_{V})=0$. Otherwise we note that
$\mathcal{U}_{V}\subseteq\\{|v_{n}|<V+1,\ \forall n>0\\}$, which is bounded
and invariant. Hence, the Poincaré recurrence theorem implies that almost
every point in $\mathcal{U}_{V}$ would return infinitely often to energy level
$|v_{n}|>V$, which is impossible as it would contradict the definition of
$\mathcal{U}_{V}.$ Our claim implies that almost all points in the energy
shell $\mathcal{W}_{V}=\\{V<v<V+1\\}$ eventually return to high energy level
$|v_{n}|>V_{*}$, which is only made possible if the ball enters the lower
chamber and the foregoing discussion ensures exponential energy growth
afterwards.

(ii) Suppose that $f_{L}(t_{1}^{*})>f_{R}(t_{1}^{*})$ and
$f_{L}(t_{2}^{*})<f_{R}(t_{2}^{*})$. Now the relative positions of the two
slits at two critical jumps indicate that the upper region is trapping and
then Proposition 4.3 guarantees an exponential energy gain at rate
$\frac{l_{1}^{+}}{l_{1}^{-}}\frac{l_{2}^{+}}{l_{2}^{-}}>1$ in the upper
chamber once the ball gets trapped. The rest of the analysis is similar to
Case (i). ∎

######  Example 5.1.

In general it is not possible to improve the result that non-escaping orbits
have finite measure to one with zero measure. For example, we start with
$\tilde{f}_{L}(t)=\tilde{f}_{R}(t)=a\cos 4\pi t+0.5$ for some small $a>0$. We
take $x_{0}=0,\lambda=0.5$, so $t_{1}^{*}=0.5,t_{2}^{*}=1.5$. We consider a
4-periodic orbit $\mathcal{P}$ starting at $t_{0}=0.25,v_{0}=2+4a$. Then

$(t_{1},v_{1})=(0.75,2+4a),\quad(t_{2},v_{2})=(1.25,2+4a),\quad(t_{3},v_{3})=(1.75,2+4a),\quad.$

We can slightly modify $\tilde{f}_{L}$ near $t_{1}^{*},t_{2}^{*}$ in such a
way that

$f_{L}(0.5)>f_{R}(0.5),\quad f_{L}(1.5)<f_{R}(1.5)$

so that the upper chamber is trapping and that the periodic orbit
$\mathcal{P}$ does not see this modification. Observe that $\mathcal{P}$ is
elliptic for all small $a$ as the trace of the collision map $F$ along
$\mathcal{P}$ is $tr(dF_{\mathcal{P}})=2-\frac{8a\pi^{2}}{1+2a}\in(0,2)$ for
$0<a<\frac{1}{4\pi^{2}-2}$. Now the matrix $dF_{\mathcal{P}}$ is conjugate to
a rotation by $2\pi\alpha$ with $\cos 2\pi\alpha=1-\frac{4a\pi^{2}}{1+2a}$. We
can easily choose $a$ such that the rotation angle $\alpha$ is Diophantine,
then Herman’s Last Geometric Theorem guarantees the stability of the elliptic
orbit $\mathcal{P}$, i.e. there exists an elliptic island of bounded
trajectories around $\mathcal{P}$ (c.f. [7, Theorem 4]).

Although the assumptions of Theorem 1 are compatible with existence of a
positive measure set of bounded orbits, we can eliminate the possibility of
oscillatory orbits. Recall that a (forward) oscillatory orbit is an orbit such
that

$\limsup_{t\to+\infty}|v(t)|=\infty\quad\text{and}\quad\liminf_{t\to+\infty}|v(t)|<\infty.$

######  Corollary 5.2.

In presence of a trapping region oscillatory orbits do not exist.

###### Proof.

We may assume without loss of generality that the lower chamber is trapping.
All the high energy orbits in the lower chamber gain energy exponentially
immediately.
Now suppose that the ball is in the upper chamber and arrives at high energy
level at some $v>V_{*}$, then it decelerates exponentially as observed in the
proof of Theorem 1 until it hits the lower chambers or the normal form no
longer applies. In either case, it either starts to accelerate exponentially
or remains in the low energy region $|v|<V_{*}$ afterwards. ∎

##  6\. Waiting Time for Exponential Acceleration

In this section we show that in the presence of the trapping region, the
majority of orbits with sufficiently high energy get trapped quickly under the
hyperbolicity assumption. Throughout this section we assume without loss of
generality that the lower chamber is trapping and $|Tr|>2$. The quantity $Tr$
is in fact the trace of the derivative of the linear map
$G_{U}=G_{UU}^{21}\circ G_{UU}^{12}$ and the hyperbolicity assumption $|Tr|>2$
indicates that $G_{U}$ is hyperbolic.

###  6.1. Almost Sure Escape for the Limiting Map

We first restrict ourselves to the linear parts $G_{UU}$’s of the dynamics in
Proposition 4.3, which approximates $P_{UU}$’s with an error of order
$\mathcal{O}(I^{-1})$ when the velocity is large $v>V_{*}$.
We note that

$\displaystyle(\bar{\tau}_{2},\bar{I}_{2})$
$\displaystyle=G_{UU}^{12}(\tau_{1},I_{1})$
$\displaystyle=\left(-\frac{l_{2}^{-}}{l_{2}^{+}}\\{\mathcal{L}_{*}\mathcal{I}_{1}(\theta_{2}^{*}-\theta_{1}^{*})-\tau_{1}\\}_{2}+1+\frac{l_{2}^{-}}{l_{2}^{+}},\frac{l_{2}^{+}}{l_{2}^{-}}\mathcal{I}_{1}+\Delta_{2}(\bar{\tau}_{2}-1)\right)$

if
$1-\frac{l_{2}^{+}}{l_{2}^{-}}<\\{\mathcal{L}_{*}\mathcal{I}_{1}(\theta_{2}^{*}-\theta_{1}^{*})-\tau_{1}\\}_{2}<1+\frac{l_{2}^{+}}{l_{2}^{-}}$.
The boundary lines

(9)
$\\{\mathcal{L}_{*}\mathcal{I}_{1}(\theta_{2}^{*}-\theta_{1}^{*})-\tau_{1}\\}_{2}=1-\frac{l_{2}^{+}}{l_{2}^{-}},\quad\\{\mathcal{L}_{*}\mathcal{I}_{1}(\theta_{2}^{*}-\theta_{1}^{*})-\tau_{1}\\}_{2}=1+\frac{l_{2}^{+}}{l_{2}^{-}}$

cut out from $R_{1}^{+}$ a sequence of boxes

$A_{n}=\\{1-\frac{l_{2}^{+}}{l_{2}^{-}}+2n<\mathcal{L}_{*}\mathcal{I}_{1}(\theta_{2}^{*}-\theta_{1}^{*})-\tau_{1}<1+\frac{l_{2}^{+}}{l_{2}^{-}}+2n\\}$

whose points will remain in the upper chamber under $G_{UU}^{12}$, while the
other points will enter the lower chamber, when jumping from right to left at
$t_{2}^{*}$.
We also observe that

$\displaystyle(\bar{\tau}_{1},\bar{I}_{1})$
$\displaystyle=G_{UU}^{21}(\tau_{2},I_{2})$
$\displaystyle=\left(-\frac{l_{1}^{-}}{l_{1}^{+}}\\{\mathcal{L}_{*}\mathcal{I}_{2}(2+\theta_{1}^{*}-\theta_{2}^{*})-\tau_{2}\\}_{2}+1+\frac{l_{1}^{-}}{l_{1}^{+}},\frac{l_{1}^{+}}{l_{1}^{-}}\mathcal{I}_{2}+\Delta_{1}(\bar{\tau}_{1}-1)\right)$

if
$1-\frac{l_{1}^{+}}{l_{1}^{-}}<\\{\mathcal{L}_{*}\mathcal{I}_{2}(2+\theta_{1}^{*}-\theta_{2}^{*})-\tau_{2}\\}_{2}<1+\frac{l_{1}^{+}}{l_{1}^{-}}$
and that the boundary lines

(10)
$\\{\mathcal{L}_{*}\mathcal{I}_{2}(2+\theta_{1}^{*}-\theta_{2}^{*})-\tau_{2}\\}_{2}=1-\frac{l_{1}^{+}}{l_{1}^{-}},\quad\\{\mathcal{L}_{*}\mathcal{I}_{2}(2+\theta_{1}^{*}-\theta_{2}^{*})-\tau_{2}\\}_{2}=1+\frac{l_{1}^{+}}{l_{1}^{-}}$

cut out from $R_{2}^{+}$ another sequence of boxes

$B_{n}=\\{1-\frac{l_{1}^{+}}{l_{1}^{-}}+2n<\mathcal{L}_{*}\mathcal{I}_{2}(2+\theta_{1}^{*}-\theta_{2}^{*})-\tau_{2}<1+\frac{l_{1}^{+}}{l_{1}^{-}}+2n\\}$

whose points will remain in the upper chamber under $G_{UU}^{21}$, while the
points outside will enter the lower chamber, when jumping from left to right
at $t_{1}^{*}$.

We define $G_{U}=G_{UU}^{21}\circ G_{UU}^{12}$ on $R_{1}^{+}$. Both
$G_{UU}^{12}$ and $G_{UU}^{21}$ are piecewise affine maps, and the derivative
of $G_{U}$ part is a constant matrix $DG_{U}=DG_{UU}^{21}\cdot DG_{UU}^{12}$
where

$DG_{UU}^{12}=\begin{pmatrix}\frac{l_{2}^{-}}{l_{2}^{+}}&-\frac{l_{2}^{-}}{l_{2}^{+}}\mathcal{L}_{*}(\theta_{2}^{*}-\theta_{1}^{*})\\\
\Delta_{2}\frac{l_{2}^{-}}{l_{2}^{+}}&-\Delta_{2}\frac{l_{2}^{-}}{l_{2}^{+}}\mathcal{L}_{*}(\theta_{2}^{*}-\theta_{1}^{*})+\frac{l_{2}^{+}}{l_{2}^{-}}\end{pmatrix},$

$DG_{UU}^{21}=\begin{pmatrix}\frac{l_{1}^{-}}{l_{1}^{+}}&-\frac{l_{1}^{-}}{l_{1}^{+}}\mathcal{L}_{*}(2+\theta_{1}^{*}-\theta_{2}^{*})\\\
\Delta_{1}\frac{l_{1}^{-}}{l_{1}^{+}}&-\Delta_{1}\frac{l_{1}^{-}}{l_{1}^{+}}\mathcal{L}_{*}(2+\theta_{1}^{*}-\theta_{2}^{*})+\frac{l_{1}^{+}}{l_{1}^{-}}\end{pmatrix}$

Since $\det(DG_{U})=1$ and $|Tr(DG_{U})|>2$, it has unstable eigenvalue
$\Lambda_{u}$ with unstable eigenvector $\bm{e}_{u}$, and stable eigenvalue
$\Lambda_{s}$ with stable eigenvector $\bm{e}_{s}$.

We observe that each box $A$ is foliated by unstable lines.
We say that an unstable line $\gamma$ in a box $A$ is good if it breaks after
one period and at least two components remain in the upper chamber, otherwise
we say it is bad.
A good line is good as a solid part of it enters the trapping region after one
period under the linear map $G_{U}$:

######  Lemma 6.1.

Let $\gamma$ be a good unstable line in some box $A$. Then the proportion of
points on $\gamma$ which remain in the upper chamber after one period is at
most

$D=\frac{1+2\frac{l_{1}^{+}}{l_{1}^{-}}}{2+\frac{l_{1}^{+}}{l_{1}^{-}}}<1.$

###### Proof.

We first note that $G_{UU}^{12}(\gamma)$ remains a complete piece in $R_{2}$
as $\gamma$ lies in $A$ and that $G_{UU}^{12}$ maps the boundaries of $A$ into
two vertical lines

$G_{UU}^{12}\left(\\{\mathcal{L}_{*}\mathcal{I}_{1}(\theta_{2}^{*}-\theta_{1}^{*})-\tau_{1}\\}_{2}=1-\frac{l_{2}^{+}}{l_{2}^{-}}\right)\subseteq\\{\tau_{2}=2\\},$
$G_{UU}^{12}\left(\\{\mathcal{L}_{*}\mathcal{I}_{1}(\theta_{2}^{*}-\theta_{1}^{*})-\tau_{1}\\}_{2}=1+\frac{l_{2}^{+}}{l_{2}^{-}}\right)\subseteq\\{\tau_{2}=0\\}.$

$G_{UU}^{12}(\gamma)$ has to stretch across at least two $B$-boxes if $\gamma$
has at least two pieces remaining in the upper chamber after one period.

2$\frac{2l_{1}^{+}}{l_{1}^{-}}$ Figure 9. A good curve partly enters the
trapping region

Suppose that $G_{UU}^{12}(\gamma)$ stretches across $N$ $B$-boxes for some
$N>1$. It is easy to see that for a fixed $N$, the highest proportion of
points staying in the upper chamber is achieved when $G_{UU}^{12}(\gamma)$
ends on the boundaries of the top and bottom boxes as shown in Figure 9.
However the height of a $B$-box is equal to $2\frac{l_{1}^{+}}{l_{1}^{-}}$
(c.f. (10)) while the height of the fundamental domain is equal to 2 (c.f.
Figure 9). This implies that in the optimal situation
$\frac{2(1-\frac{l_{1}^{+}}{l_{1}^{-}})(N-1)}{2N+2\frac{l_{1}^{+}}{l_{1}^{-}}}$
of the points on $G_{UU}^{12}(\gamma)$ land in the lower chamber after jumping
from left to right at $t_{1}^{*}$. This proportion is larger than
$\frac{(1-\frac{l_{1}^{+}}{l_{1}^{-}})(2-1)}{2+\frac{l_{1}^{+}}{l_{1}^{-}}}$
as it is an increasing function in $N$ and $N\geq 2$. Then the largest portion
which remains in the upper chamber is given by
$D=\frac{1+2\frac{l_{1}^{+}}{l_{1}^{-}}}{2+\frac{l_{1}^{+}}{l_{1}^{-}}}.$
Recall that the relative positions of two slits at $t_{1}^{*}$ implies that
$\frac{l_{1}^{+}}{l_{1}^{-}}<1$, so $D<1$. ∎

Next we need to control the number of the short bad pieces as an unstable line
breaks under the iterations of the linear map $G_{U}$.
Suppose $\gamma$ is an unstable line in some box $A$. For $x\in\gamma$, we
denote as $r_{n}(x)$ the distance from $x_{n}$ to the nearest boundary of the
component $\gamma_{n}$ containing $x_{n}$. Employing the argument in Section 5
of [4] we obtain the following Growth Lemma.

######  Lemma 6.2 (Growth Lemma).

There exists a constant $C^{*}$ s.t. for any small $\epsilon>0$ and any
$n\in\mathbb{N}$

${\mathrm{mes}}_{\gamma}\\{x\in\gamma:r_{n}(x)<\epsilon\\}\leq C^{*}\epsilon$

###### Proof.

Let $k_{n}(\delta)$ denote the max number of the pieces that an unstable line
of length less than $\delta$ can be cut into. We define $k_{n}=\lim_{\delta\to
0}k_{n}(\delta)$. We claim that $k_{n}\leq 8n$. Indeed since the singularities
of $G_{U}^{n}$ are lines and there are at most $8n$ possibilities for slopes.
Consequently, there exists $\delta_{0}$ so small that $k_{n}(\delta)\leq 16n$
for any $\delta<\delta_{0}$. We choose $n_{0}$ such that
$\frac{32n_{0}}{\Lambda_{u}^{n_{0}}}<1$ and by replacing $G_{U}$ with
$G_{U}^{n_{0}}$ we can always assume $n_{0}=1$.
For inductive purposes we cut a long unstable line into pieces shorter than
$\delta_{0}$ and let $\bar{r}_{n}(x)$ denote the distance from $x_{n}$ to the
nearest real or artificial boundary of the component containing $x_{n}$. We
note that by doing so we improve the estimate as $\bar{r}_{n}(x)\leq r_{n}(x)$
and it suffices to prove the statement for $\bar{r}_{n}$.
First we observe that

(11)
${\mathrm{mes}}_{\gamma}\\{\bar{r}_{0}(x)<\epsilon\\}\leq\frac{2L}{\delta_{0}}\epsilon.$

$\bar{r}_{n+1}(x)$ is less than $\epsilon$ if $x_{n+1}$ either passes a real
or artificial singularity, where $L$ is the unstable height of $\gamma$. The
former is controlled by
$2k_{1}(\delta_{0}){\mathrm{mes}}_{\gamma}\\{r_{n}<\frac{\epsilon}{\Lambda_{u}}\\}$
while the latter by $2k_{1}(\delta_{0})\frac{L}{\delta_{0}}\epsilon$.
Therefore

${\mathrm{mes}}_{\gamma}\\{\bar{r}_{n+1}<\epsilon\\}\leq\frac{32}{\Lambda_{u}}{\mathrm{mes}}_{\gamma}\\{r_{n}<\epsilon\\}+32\frac{L}{\delta_{0}}\epsilon.$

Thus by induction we conclude that

${\mathrm{mes}}_{\gamma}\\{\bar{r}_{n}<\epsilon\\}\leq\left(\frac{32}{\Lambda_{u}}\right)^{n}{\mathrm{mes}}_{\gamma}\\{\bar{r}_{0}(x)<\epsilon\\}+32\frac{L}{\delta_{0}}\epsilon\left(1+\cdots+\left(\frac{32}{\Lambda_{u}}\right)^{n-1}\right).$

Since $\frac{32}{\Lambda_{u}}<1,$ (11) gives the desired growth control with

$C^{*}=\left(\frac{32}{\Lambda_{u}}\right)\frac{2L}{\delta_{0}}+\frac{32L}{\delta_{0}(1-\frac{32}{\Lambda_{u}})}.\qed$

Finally we show that under the linear approximation map $G_{U}$ almost every
point will eventually escape to the trapping region:

######  Proposition 6.3.

In each box $A$, for any $\epsilon>0$, there exists $N=N(\epsilon)$ such that
all but an $\epsilon$-measure set of points in $A$ enter the lower chamber
within $N$ periods. In particular, almost every point will leave the upper
chamber in the future.

###### Proof.

Fix $\epsilon>0$. Choose $k,l$ such that

$D^{k}L<0.5\epsilon\quad\text{and}\quad(kl+1)\frac{C^{*}+L^{2}}{\Lambda_{u}^{l/2}}<0.5\epsilon,$

and take $N=kl+1$.
We suppose that under the linear map $G_{U}$ a point $x$ on a unstable line
$\gamma$ stays in the upper chamber up to $N$ periods.
If the trajectory of $x$ lands on good lines more than $k$ times in $N$
periods, then Lemma 6.1 shows that for good lines the portion which remains in
the upper chamber in the next period is at most $D.$ Hence by induction we see
that

${\mathrm{mes}}_{\gamma}\\{x\in\gamma:\hbox{$\\{P_{U}^{m}x\\}_{m=0}^{N}$
visits good lines more than $k$ times}\\}<D^{k}L$

If instead the trajectory of $x$ visits good lines less than $k$ times in $N$
periods, then it has to visit consecutively $l$ bad lines at least once in $N$
periods.
Now suppose that the trajectory segment $x_{n},\cdots,x_{n+l}$ land on bad
lines $\gamma_{n},\cdots,\gamma_{n+l}$ for some $n<N$ and we denote as
$\mathcal{B}_{n}$ the set of all such $x\in\gamma$ that lands badly during $n$
to $n+l$ periods. We subdivide $\mathcal{B}_{n}$ into $\mathcal{B}_{n,L}$ and
$\mathcal{B}_{n,S}$, where $\mathcal{B}_{n,L}$ collects points with
$|\gamma_{n}|\geq\Lambda_{u}^{-l/2}$ and $\mathcal{B}_{n,S}$ collects points
with $|\gamma_{n}|<\Lambda_{u}^{-l/2}$.
By Lemma 6.2, $|\mathcal{B}_{n,S}|\leq C^{*}\Lambda_{u}^{-l/2}$. On the other
hand, it follows from uniform hyperbolicity that

${\mathrm{mes}}_{\gamma_{n}}\\{\hbox{$x_{n}$ returns badly for next $l$
periods}\\}=\frac{|\gamma_{n+l}|}{\Lambda_{u}^{l}}\leq\frac{L}{\Lambda_{u}^{l}}.$

Hence

$|\mathcal{B}_{n,L}|\leq\frac{L}{\Lambda_{u}^{l}}\sum_{|\gamma_{n}|\geq\Lambda_{u}^{-l/2}}|G_{U}^{-n}\gamma_{n}|\leq\frac{L}{\Lambda_{u}^{l}}|\gamma|\leq\frac{L^{2}}{\Lambda_{u}^{l}}.$

Combining the estimates on $\mathcal{B}_{n,L}$ and $\mathcal{B}_{n,S}$, we
have

$|\mathcal{B}_{n}|\leq\frac{C^{*}}{\Lambda_{u}^{l/2}}+\frac{L^{2}}{\Lambda_{u}^{l}}\leq\frac{C^{*}+L^{2}}{\Lambda_{u}^{l/2}}.$

Consequently the set $\mathcal{B}$ of points on $\gamma$ which make $l$
consecutive bad landings is controlled in size by

$|\mathcal{B}|\leq(kl+1)\frac{C^{*}+L^{2}}{\Lambda_{u}^{l/2}}.$

Since a box $A$ is foliated by unstable lines, we conclude by a disintegration
of measure argument and our choice of $k,l$ that under the linear map $G_{U}$
the set of points in $A$ which stay in the upper chamber for at least $N$
periods has measure less than $\epsilon.$ ∎

###  6.2. Quick Escape for the Actual Map.

By Proposition 4.3, the fundamental domains of $P_{UU}$ are
$\mathcal{O}(\mathcal{I}^{-1})$-deformation of the boxes $A,B$ and
$P_{UU}=G_{UU}+\mathcal{O}(\mathcal{I}^{-1})$.
Now we prove Theorem 2.

###### Proof of Theorem 2.

Fix $\epsilon>0$ and a box $A$ with large energy $\tilde{\mathcal{I}}_{0}$ (to
be specified later). By Proposition 6.3 we choose $N$ such that in each
$A$-box the points that remain in the upper chamber up to $N$ periods under
the linear approximation $G_{U}$ take up a set of measure less than
$0.5\epsilon$, i.e. we take $N=kl+1$ where $k,l$ are integers such that

(12) $k>\frac{\log(0.25\epsilon/L)}{\log
D}\quad\text{and}\quad\frac{kl+1}{\Lambda_{u}^{l/2}}<\frac{0.25\epsilon}{C^{*}+L^{2}}.$

We shall show that the statement of Theorem 2 holds with some large
$\tilde{\mathcal{I}}_{0}=\tilde{\mathcal{I}}_{0}(\epsilon)$ and

(13) $T=2N.$

Let $A_{n}^{\delta}$ and $B_{n}^{\delta}$ denote the points in $A_{n}$ and
$B_{n}$ which are closer than $\delta$ to the boundary,

$\mathcal{A}^{\delta}=\bigcup_{n}A_{n}^{\delta},\quad\widetilde{\mathcal{A}}^{\delta}=\bigcup_{n}\left(A_{n}-\setminus
A_{n}^{\delta}\right),$
$\mathcal{B}^{\delta}=\bigcup_{n}B_{n}^{\delta},\quad\widetilde{\mathcal{B}}^{\delta}=\bigcup_{n}\left(B_{n}-\setminus
B_{n}^{\delta}\right).$

Choose $\delta<0.5\epsilon$ so that the set of points in the box $A$ which
visit either $\mathcal{A}^{\delta}$ or $\mathcal{B}^{\delta}$ during the first
$N$ iterations is less than $0.5\epsilon.$

By Proposition 4.3, there is a constant $C_{1}$ such that if
$P_{UU}^{12}(x)\in\widetilde{\mathcal{B}}^{C_{1}/\mathcal{I}}$ and
$P_{UU}^{21}(P_{UU}^{12}x)\in\widetilde{\mathcal{A}}^{C_{1}/\mathcal{I}}$ then
the orbit of $x$ stays in the upper chamber for the next period and

$\left|P_{U}(x)-G_{U}(x)\right|\leq\frac{C_{1}}{\mathcal{I}}\quad\text{where}\quad
P_{U}=P_{UU}^{21}\circ P_{UU}^{12}.$

Accordingly there is a constant $C_{2}$ such that if for some $n\leq N$

(14)
$P_{UU}^{12}P_{U}^{k}(x)\in\widetilde{\mathcal{B}}^{\frac{C_{2}\Lambda_{u}^{N}}{\mathcal{I}^{*}}},\quad
P_{U}^{k+1}(x)\in\widetilde{\mathcal{A}}^{\frac{C_{2}\Lambda_{u}^{N}}{\mathcal{I}^{*}}}$

and $\mathcal{I}_{k}\geq\mathcal{I}^{*}$ for $k<n$ then the real orbit of $x$
stays in upper chamber for the first $n$ iterations and

(15)
$\left|P_{U}^{n}(x)-G_{U}^{n}(x)\right|\leq\frac{C_{2}\Lambda_{u}^{N}}{\mathcal{I}^{*}}.$

Next, set $C_{3}=\frac{l_{2}^{+}l_{1}^{+}}{l_{2}^{-}l_{1}^{-}}<1.$ Then during
$N$ iterations the value of $I$ cannot drop by more than $C_{3}^{N}$ times.
Hence if $x$ satisfies (14) and $\mathcal{I}_{0}\geq C_{3}^{N}\mathcal{I}^{*}$
then (15) holds.

Now choose $\mathcal{I}^{*}$ so that

(16) $\frac{C_{2}\Lambda_{u}^{N}}{\mathcal{I^{*}}}<\delta$

Now we consider the orbits where
$\bar{\mathcal{I}}\leq\mathcal{I}_{0}\leq\bar{\mathcal{I}}+1$ for some
$\bar{\mathcal{I}}\geq C_{3}^{N}\mathcal{I}^{*}.$ There are three
possibilities:

(i) The real orbit of $x$ leaves the upper chamber at some period $n<N$;

(ii) The real orbit of $x$ stays in $\widetilde{\mathcal{A}}^{\delta}$ for the
first $N$ iterations;

(iii) The real orbit of $x$ stays in $\widetilde{\mathcal{A}}^{\delta}$ until
it hits $\mathcal{A}^{\delta}\cup(P_{U}^{12})^{-1}\mathcal{B}^{\delta}$ at
some period $n<N$.

Proposition 6.3 and our choice of $\delta$ and $\mathcal{I}^{*}$ imply that
the set of orbits where either (ii) or (iii) happens has measure smaller than
$\epsilon.$

This completes the proof of Theorem 2. ∎

##  7\. Conclusion

We have described in this paper a two-dimensional exponential Fermi
accelerator: a rectangular billiard with two moving slits. We found a
mechanism for a particle to gain energy exponentially fast, i.e. the trapping
regions. When the relative positions of two slits change at two critical
jumps, a trapping region, either the upper or lower chamber, is created so
that every high velocity orbit starts to gain energy exponentially fast once
it gets trapped. We demonstrated that a trapping region exists for sizable
choices of parameters and the exponential acceleration happens for almost all
high energy orbits. Moreover under additional hyperbolicity assumptions on the
parameters we provided an explicit estimate on the waiting time until which
the exponential acceleration starts for most high-energy orbits.

It is worth noting that all the analysis done in this paper is based on the
normal forms in the high energy region. The normal form implies exponential
energy growth almost surely if a particle starts with sufficiently high
initial velocity, and it eliminates the possibility of oscillatory orbits. The
normal forms do not apply in low energy region where we might have bounded
orbits for certain wall motion. In this paper we did not analyze the case when
a trapping region does not exist, which can be easily achieved by choosing
parameters such that the relative positions of two slits do not change at two
critical jumps. Our normal forms still apply even in this complicated case but
the analysis would be more delicate as the particle needs to make a choice of
traveling up or down every time it jumps. We also note that in the non-
resonant case when the periods of the particle and the wall are
incommensurable, the normal form still applies, however the jumping time
depends on the period. In particular, the jumping times become dense on the
period which precludes the existence of the trapping region, so the problem
becomes similar to the resonant non-trapping case. These observations provide
possible directions for future work.

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