A Rectangular Billiard with Moving Slits
Jing Zhou
math.DS math-ph math.MP




A Rectangular Billiard with Moving Slits
    Jing Zhou
    December 30, 2023
========================================





    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.







§ INTRODUCTION

In an attempt to explain the existence of high energy particles in cosmic rays, Fermi <cit.> in 1949 proposed a model in which charged particles undergo repeated reflections in moving magnetic fields. 
Later in 1961 Ulam <cit.> 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 <cit.> 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 <cit.> 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 <cit.> <cit.> <cit.>, 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 <cit.> 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 <cit.> 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 <cit.>
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 
<cit.>.



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 <cit.>. In a Lorentz gas model <cit.>, 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 <cit.> 
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
<cit.>.
The models which are closest to our setting are the following.
 Shah, Turaev and Rom-Kedar <cit.> 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 <cit.> 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 <cit.>. 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 λ and the length of the right slit 1-λ, where λ∈ (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.






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≤ x_0 < λ. The ball jumps from the left slit to right one at time t_1^*=λ-x_0 and then from right to the left at time t_2^*=2-λ-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=λ. 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.










§ MAIN RESULTS

In this section we describe the exponential accelerator we have found.

For a wide range of choices in λ 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 <ref>). A trapping region is created in this case and the ball starts to gain energy exponentially fast once it gets trapped.



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_* ≫ 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) = 
       
         f_L(t)     0<t<t_1^*  or  t_2^*<t<2 
    
         f_R(t)     t_1^*<t<t_2^*.


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

    Tr^U    = ( 1-f_1^-/1-f_1^+ - a_1 β)( 1-f_2^-/1-f_2^+ - a_2 α) + ( 1-f_1^+/1-f_1^- - a_1 α)( 1-f_2^+/1-f_2^- - a_2 β)
            - a_1 a_2 αβ

where f_i^±=f(t_i^* ±), a_i=ḟ_i^-(1-f_i^+)-ḟ_i^+(1-f_i^-), 

    α=(∫_0^t_1^*+∫_t_2^*^2) ds/(1-f(s))^2  and  β= ∫_t_1^*^t_2^*ds/(1-f(s))^2.
 
If the upper chamber is trapping, then we define in the lower chamber 

    Tr^L = ( f_1^-/f_1^+ - a_1' β' )( f_2^-/f_2^+ - a_2' α' ) + ( f_1^+/f_1^- - a_1' α' )( f_2^+/f_2^- - a_2' β' ) - a_1' a_2' α' β'

where a_i'=ḟ_i^+ f_i^- - ḟ_i^- f_i^+, 
α'=(∫_0^t_1^*+∫_t_2^*^2) ds/f(s)^2 and β'= ∫_t_1^*^t_2^*ds/f(s)^2.



 Assume λ 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, ζ>0 such that for any ϵ>0, there exists V_0=V_0(ϵ) and T=T(ϵ) such that for each V≥ V_0 the complement of set 
    {(t_0,v_0): |v_0|∈ [V, V+1]: ∀ t≥ T   |v(t)|≥|v(T)|/K e^ζ t }
 has measure less than ϵ, 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 ϵ
(see equations (<ref>) and (<ref>)).







 To illustrate our results, we consider the case where

    f_L(t)=0.3cos(π t)+0.5,   f_R(t)=0.3sin(π t)+0.5.

Then 

    Δ(t):=f_L(t)-f_R(t)=0.3 √(2)cos(π t+π/4).

A trapping region exists for λ, x_0 such that either 

    Δ(λ-x_0)>0,  Δ(2-λ-x_0)<0  or  Δ(λ-x_0)<0,  Δ(2-λ-x_0)>0.

The former case is equivalent to λ-x_0<0.25, 0.75<λ+x_0<1.75; the upper chamber is trapping and the hyperbolicity assumption holds if |Tr^L|>2.
 
The latter case is equivalent to λ-x_0>0.25, λ+x_0<0.75;
the lower chamber is trapping and the hyperbolicity assumption holds if 
|Tr^U|>2.
 
Figure <ref> demonstrates that for f_L, f_R defined above by (<ref>)
the assumptions of Theorems <ref> and <ref> hold for a sizable set of parameters.


The structure of the rest of the paper is the following.
In Section <ref> we describe the collision map.
In Section <ref> 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 <ref> is given in Section <ref> and
the proof of Theorem <ref> is given in Section <ref>.
In Section <ref> we summarize the tools developed
in the present paper and discuss open problems.











§ 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 
𝒮_i = { t=t_i^* } and its image F𝒮_i. Also let R̃_i = F^-1R_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) ∈R̃ 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. <ref> on the left).

Two consecutive collisions (t_n,v_n) and (t_n+1,v_n+1)=F(t_n,v_n) satisfy 

    {   v_n+1 = v_n +2ḟ(t_n+1)
       2-f(t_n) -f(t_n+1) = v_n(t_n+1-t_n).
      .






Similarly, suppose that the ball is initially in the lower chamber. If for (t,v) ∈R̃ 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. <ref> on the right).
Two consecutive collisions satisfy 

    {   v_n+1 = v_n +2ḟ(t_n+1)
       f(t_n) + f(t_n+1) = -v_n(t_n+1-t_n).
      .



Now let us examine the switching cases.


Suppose that the ball is initially in the upper chamber and two consecutive collisions still follow <Ref> 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) ∈R̃, if v(t_* -t) > 2-f(t)-f(t_*+), then the ball collides with 
the ceiling and then enters the lower chamber (c.f. <Ref> 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. <Ref> on the right).





Two consecutive collisions satisfy the following <Ref> in the first case

    {   v_n(t_n+1-t_n) =f(t_n+1) -f(t_n) +2
       v_n+1 = -v_n+2ḟ(t_n+1)
      .


and the following <Ref> in the second case


    {   v_n(t_n+1-t) = f(t_n+1) -f(t_n)
       v_n+1 = -v_n+2ḟ(t_n+1).
      .




On the other hand, suppose the ball is initially in the lower chamber and two consecutive collisions still follow <Ref> 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) ∈R̃, if -v(t_* -t) > f(t)+f(t_*-), then the ball 
collides with the floor and enters the upper chamber (c.f. <Ref> 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. <Ref> on the right).





Two consecutive collisions satisfy the following <Ref> in the first case

    {   v_n(t_n+1-t_n) =f(t_n+1) -f(t_n) -2
       v_n+1 = -v_n+2ḟ(t_n+1)
      .


and the following <Ref> in the second case


    {   v_n(t_n+1-t) = f(t_n+1) -f(t_n)
       v_n+1 = -v_n+2ḟ(t_n+1).
      .











§ THE NORMAL FORM


In this section we study how the (vertical) velocity of the ball changes after one period Δ 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.




 §.§ 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 ℒ_*=∫_0^2 l^-2(s) ds.


For (t,v)∉ R_i ∪R̃_i (i=1,2) and v≫ 1, there exists an action-angle coordinate (θ,I) = Ψ_U (t,v) ∈ℝ/2ℤ×ℝ_+ such that 

    θ_n+1 = θ_n + 2/I_n + 𝒪( 1/I_n^4),  I_n+1 = I_n + 𝒪(1/I_n^3).

In fact, θ =θ(t) =2/ℒ_*∫_0^t ds/l(s)^2 2, I = I(t,v) = ℒ_*/2(lv +ll̇ + l^2l̈/3v).


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

    v_n+1-v_n ≈ -2l̇(t_n),   t_n+1-t_n ≈2l(t_n)/v_n.

This leads to the Euler scheme of the following ODE 

    dv/dt = -vl̇/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 I=lv +ll̇ + l^2l̈/3v.

Next, the formula for θ can be obtained reversely by solving the ODE 
    θ'2l/v=2/lv

which leads to θ(t) = ∫_0^t l^-2(s) ds.

We observe that only the order v term in I is used to derive the formula for θ and it seems to produce an estimate only up to first order 

    θ_n+1 - θ_n = 2/I_n + 𝒪(I_n^-2).
 
But in fact by noting the Taylor expansion of l^-2 and that

    t_n+1 - t_n = 2l(t_n)/v_n + 2l(t_n)l̇(t_n)/v_n^2 + 2l(t_n)l̇(t_n)^2 + 2l(t_n)^2l̈(t_n)/v_n^3 + 𝒪(v_n^-4)

we obtain that 

    ∫_t_n^t_n+1 l^-2(s) ds 
       = t_n+1 - t_n/l_n^2 - l̇_n/l_n^3(t_n+1 - t_n)^2 + (l̇_n^2/l_n^4-l̈_n/3l_n^3)(t_n+1 - t_n)^3 + 𝒪(v_n^-4)
       = 2/l_nv_n - 2l̇_n/t_nv_n^2 +2l̇_n^2-2/3l_nl̈_n/l_nv_n^3 +𝒪(v_n^-4)
       = 2/I_n + 2l̇_n/v_nI_n +2l_nl̈_n/3v_n^2I_n- 2l̇_n/l_nv_n^2 + 2l̇_n^2-2/3l_nl̈_n/l_nv_n^3+𝒪(v_n^-4)
       =2/I_n + 2l̇_n^2/v_n^2 I_n + 2l_nl̈_n/3v_n^2 I_n + 2l̇_n^2-2/3l_nl̈_n/l_nv_n^3 + 𝒪(v_n^-4)
       =2/I_n + 𝒪(v_n^-4)

where l_n = l(t_n) and l̇_n = l̇(t_n), which produces the desired third order estimate.


Finally, we need to rescale θ (and hence I) to make θ 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 <Ref> becomes the same as <Ref> with g in place of f 

    {   v_n+1 = v_n +2ġ(t_n+1)
       2-g(t_n) -g(t_n+1) = v_n(t_n+1-t_n)
      .

Therefore all the computation above in Lemma <ref> applies with g in the place of f.

We define m(t) = 1-g(t) = -f(t) and ℳ_*= ∫_0^2 m(s)^-2 ds.
We have an action-angle coordinate if the collision occurs in the lower chamber away from singularities 


For (t,v)∉ R_i ∪R̃_i (i=1,2) and v≪ -1, there exists an action-angle coordinate (ζ,J) = Ψ_L (t,v) ∈ℝ/2ℤ×ℝ_+ such that 

    ζ_n+1 = ζ_n + 2/J_n + 𝒪( 1/J_n^4),  J_n+1 = J_n + 𝒪(1/J_n^3).

In fact, ζ =ζ(t) =2/ℳ_*∫_0^t ds/m(s)^2 2, J = J(t,v) = ℳ_*/2(mv +mṁ + m^2m̈/3v).







 §.§ 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.




  §.§.§ 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 <ref> 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.





For (t_1,v_1) ∈ R_1^+ with v_1 > V_*, we denote (t̃_2,ṽ_2) =F^n_1(t_1,v_1) ∈R̃_2^+, where n_1 = [I_1/2(θ_2^* - θ_1)] and θ_2^* = 2/ℒ_*∫_0^t_2^*ds/l(s)^2, and (t̅_2,v̅_2) = F(t̃_2,ṽ_2) ∈ R_2^+. Similarly, for (t_2,v_2) ∈ R_2^+ with v_2 ≫ 1, we denote (t̃_1,ṽ_1) =F^n_2(t_2,v_2) ∈R̃_1^+, where n_2 = [I_2/2(2+θ_1^* - θ_2)] and θ_1^* = 2/ℒ_*∫_0^t_1^*ds/l(s)^2, and (t̅_1,v̅_1) = F(t̃_1,ṽ_1) ∈ R_1^+.

We introduce a new pair of variables (τ, ℐ) defined on the upper singular strips  

    τ =
    
    I(θ - θ_1^*)    on R_1^+
    
    I(θ - θ_2^*)    on R_2^+, ℐ=I/ℒ_*  on R_1^+, R_2^+


Now we present the Poincaré maps P^12_UU:R_1^+ → R_2^+ and P^21_UU:R_2^+ → 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):

    Δ_i = 1/2l(t_i^*+)/l(t_i^*-)( l(t_i^*-)l̇(t_i^*+) - l(t_i^*+)l̇(t_i^*-) ),
       Δ'_i = 1/8 l(t_i^*+)^2 ( l(t_i^*-)l̈(t_i^*+) - l(t_i^*+)l̈(t_i^*-) ),
       Δ”_i = 1/24 l(t_i^*-)l(t_i^*+) ( l(t_i^*-)l̈(t_i^*+) - l(t_i^*+)l̈(t_i^*-) ).




Suppose that (τ_1,ℐ_1) ∈ R_1^+ and ℐ_1>V_*, and that 

    f(t^*_2+)-f(t^*_2-) ≲ l_2^- {ℒ_*ℐ_1(θ_2^* - θ_1^*) -τ_1 }_2 ≲ 2-f(t^*_2+)-f(t^*_2-),
 
where ≲ means the inequality holds up to an error of order 𝒪(1/ℐ), 
and {∙}_2 = ∙ 2. Then the Poincaré map P_UU^12: R_1^+ → R_2^+ is given by (τ̅_2,ℐ̅_2) = G_UU^12(τ_1,ℐ_1) + H_UU^12(τ_1,ℐ_1) + 𝒪(ℐ_1^-2) where 

    G_UU^12(τ_1,ℐ_1) =( -l_2^-/l_2^+{ℒ_*ℐ_1(θ_2^* - θ_1^*) -τ_1 }_2 + 1+ l_2^-/l_2^+, l_2^+/l_2^-ℐ_1 + Δ_2 (τ̅_2 -1) )

and

    H_UU^12(τ_1,ℐ_1) =(0, Δ'_2 (τ̅_2 -1)^2/ℐ_1 + Δ”_2/ℐ_1 )

Similarly, suppose that (τ_2,ℐ_2) ∈ R_2^+, ℐ_2 > V_*, and that 

    f(t^*_1+)-f(t^*_1-) ≲ l_1^- {ℒ_*ℐ_2(2+θ_1^* - θ_2^*) -τ_2 }_2  ≲ 2-f(t^*_1+)-f(t^*_1-).
 
Then the Poincaré map P_UU^21: R_2^+ → R_1^+ is given by 

    (τ̅_1,ℐ̅_1) = G_UU^21(τ_2,ℐ_2) + H_UU^21(τ_2,ℐ_2) + 𝒪(ℐ_2^-2)
 
where 

    G_UU^21(τ_2,ℐ_2) =( -l_1^-/l_1^+{ℒ_*ℐ_2(2+θ_1^* - θ_2^*) -τ_2 }_2 + 1+ l_1^-/l_1^+, l_1^+/l_1^-ℐ_2 + Δ_1 (τ̅_1 -1) ),
 

    H_UU^21(τ_2,ℐ_2) =(0, Δ'_1 (τ̅_1 -1)^2/ℐ_2 + Δ”_1/ℐ_2 )

and l_i^± = l(t_i^* ±).




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^*

    l(t̃^*) = l_-(t^*) + l̇_-(t̃-t^*) + 1/2l̈_-(t^*)(t̃-t^*)^2 + 𝒪(ṽ^-3),
       l(t̃^*) = l_+(t^*) + l̇_+(t̅-t^*) + 1/2l̈_+(t^*)(t̅-t^*)^2 + 𝒪(ṽ^-3),
       l̇(t̃^*) = l̇_-(t^*) + l̈_-(t̃-t^*) + 𝒪(ṽ^-2),
       l̇(t̃^*) = l̇_+(t^*) + l̈_+(t̅-t^*) + 𝒪(ṽ^-2),
       l̈(t̃^*) = l̈_-(t^*) +𝒪(ṽ^-1),
       l̈(t̃^*) = l̈_+(t^*) +𝒪(ṽ^-1),

and that 

    v̅ = ṽ - 2l̇(t̅),   t̅-t̃ = l(t̅)+l(t̃)/v̅.

Hence by solving iteratively the implicit equation we attain

    t̅-t̃=l_+ +l_-/ṽ + (l̇_+ +l̇_-)t̅-t^*/ṽ - l̇_-(l_+ +l_-)/ṽ^2 + 𝒪(ṽ^-3).

By a straightforward but tedious computation we arrive at

    2ℒ_*^-1( l_-/l_+I̅-Ĩ) 
       = (l_+l̇_- - l_-l̇_+) - l_+l̇_- - l_-l̇_+/l_+(t̅ - t^*)v̅
           +(l_+ l̈_- - l_- l̈_+ + l_-/l_+l̇_+^2 - l̇_-l̇_+)(t̅ - t^*)
           +(l_-/3(l_+ l̈_+ - l_- l̈_-) + l̈_-/2(l_-^2 - l_+^2))1/v̅
           - l_+ l̈_- - l_- l̈_+/2l_+(t̅ - t^*)^2v̅ + 𝒪(ṽ^-2)
       = l_-l̇_+ - l_+l̇_-/l_+( (t̅ - t^*)v̅(1+l̇_+/v̅) - l_+ )
           + l_- l̈_+ - l_+ l̈_-/2l_+ v̅( ((t̅ - t^*)v̅- l_+ )^2 + l_-l_+(l^- l̈_- - l_+ l̈_+)/3(l_- l̈_+ - l_+ l̈_-)) + 𝒪(ṽ^-2).

It can be checked directly by Taylor expanding I̅ and θ̅ that 

    τ̅ = I̅(θ̅ - θ_2^*) = 1/l_+( (t̅-t^*)v̅ + l̇_+ (t̅-t^*) ) + 𝒪(ṽ^-2).

Thus eventually we have

    ℐ̅ =  (l_+/l_-) ℐ̃ + Δ (τ̅ -1) + Δ' (τ̅ -1)^2/ℐ̃ + Δ”/ℐ̃ + 𝒪(ℐ̃^-2).

Now we compute τ̅. Observe that 

    τ̅ = v̅ + l̇_+ /l_+(t̅-t^*) + 𝒪(v̅^-2), Ĩ(θ̃ - θ̃_̃2̃^̃*̃) = ṽ + l̇_- /l_-(t̃-t^*) + 𝒪(v̅^-2).

Therefore 

    l_+ τ̅   = ((t̅-t̃) + (t̃-t^*))(ṽ+l̇_- - (l̇_- + l̇_+)) + 𝒪(ṽ^-2)
       = (t̃-t^*)(ṽ+l̇_-) + (t̅-t̃)(ṽ+l̇_-) - (t̅-t^*)(l̇_- + l̇_+) + 𝒪(ṽ^-2)
       = l_-Ĩ(θ̃ - θ̃_̃2̃^̃*̃) 
    + l_- + l_+/ṽ (ṽ+l̇_-) 
    +(l̇_- + l̇_+)(t̅-t^*) 
       - l̇_-(l_- + l_+)/ṽ - (t̅-t^*)(l̇_- + l̇_+) + 𝒪(ṽ^-2)
       = l_-Ĩ(θ̃ - θ̃_̃2̃^̃*̃) + l_- + l_+ + 𝒪(ṽ^-2),

which gives 

    τ̅ = (l_-/l_+) Ĩ(θ̃ - θ_2^*) + 1+ l_-/l_+ + 𝒪(Ĩ^-2).

But <Ref> implies that 

    Ĩ = I + 𝒪(I^-2),  θ̃ =θ + 2n_1/I + 𝒪(I^-3)

hence we have 

    Ĩ(θ̃ - θ_2^*)= τ + 2n_1 + I(θ_1^* - θ_2^*) + 𝒪(I^-2)

where n_1=[I_1/2(θ_2^*-θ_1)].

We hitherto complete the proof of the formula for P_UU^12.






  §.§.§ 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)

    ζ_i^*   =2/M_*∫_0^t_i^* m(s)^-2 ds
    Υ_i   =1/2m_i^+/m_i^-(m_i^-ṁ_i^+ - m_i^+ṁ_i^-)
    Υ_i'   =1/8m_i^+2(m_i^-m̈_i^+ - m_i^+m̈_i^-)
    Υ_i”   =1/24m_i^-m_i^+(m_i^-m̈_i^- - m_i^+m̈_i^+)


We introduce a new pair of variables (ρ,𝒥) on the lower singular strips, which is the counterpart of (τ,ℐ) as follows 

    ρ =
    
    J(ζ - ζ_1^*)    on R_1^-
    
    J(ζ - ζ_2^*)    on R_2^- ,  𝒥=J/ℳ_* on R_1^-,R_2^-




Suppose that (ρ_1,𝒥_1) ∈ R_1^-, and 𝒥_1 > V_*, and that 

    f(t^*_2-)-f(t^*_2+) ≲ -m_2^-{ℳ_*𝒥_1(ζ_2^* - ζ_1^*) -ρ_1 }_2 ≲ f(t^*_2-)+f(t^*_2+).
 
Then the Poincaré map P_LL^12 : R_1^- → R_2^- is given by 

    (ρ̅_2,𝒥̅_2) = G_LL^12(ρ_1,𝒥_1) + H_LL^12(ρ_1,𝒥_1)+ + 𝒪(𝒥_1^-2)
 where

    G_LL^12(ρ_1,𝒥_1)=( -m_2^-/m_2^+{ℳ_*𝒥_1(ζ_2^* - ζ_1^*) -ρ_1 }_2 + 1+ m_2^-/m_2^+, m_2^+/m_2^-𝒥_1 + Υ_2 (ρ̅_2 -1) )

and 

    H_LL^12(ρ_1,𝒥_1) = (0, Υ'_2 (ρ̅_2 -1)^2/𝒥_1 + Υ”_2/𝒥_1 ).

Similarly, suppose that (ρ_2,𝒥_2) ∈ R_2^- and 𝒥_2 > V_*, and that 

    f(t^*_1-)-f(t^*_1+) ≲ -m_1^-{ℳ_*𝒥_2(2+ζ_1^* - ζ_2^*) -ρ_2 }_2 ≲ f(t^*_1-)+f(t^*_1+).
 
Then the Poincaré map P_LL^21 : R_2^- → R_1^- is given by 

    (ρ̅_1,𝒥̅_1) = G_LL^21(ρ_2,𝒥_2) + H_LL^21(ρ_2,𝒥_2)+ + 𝒪(𝒥_2^-2)
 
where

    G_LL^21(ρ_2,𝒥_2)=( -m_1^-/m_1^+{ℳ_*𝒥_2(2+ζ_1^* - ζ_2^*) -ρ_2 }_2 + 1+ m_1^-/m_1^+, m_1^+/m_1^-𝒥_2 + Υ_1 (ρ̅_1 -1) ),



    H_LL^21(ρ_2,𝒥_2) = (0, Υ'_1 (ρ̅_1 -1)^2/𝒥_2 + Υ”_1/𝒥_2 ),

and m_i^± = m(t_i^* ±).







  §.§.§ The Upper-Lower Chamber Case

Now we suppose that (t̃,ṽ) ∈R̃ 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 (t̅,v̅)=F(t̃,ṽ).



If v(t_* -t) > 2-f(t)-f(t_*+), then the ball collides 
with the ceiling and then enters the lower chamber (c.f. <Ref> 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-ṽ(t_*-t̃)+l(t̃), so that the two consecutive collisions at the moving slits are concatenated by two fictional collisions at the imaginary wall, to which the prop43 and prop44 formulas readily apply.




More precisely, as the ball leaves the previous slit at time t̃ with velocity ṽ, it collides against the imaginary tiny slit at time t_* and the outgoing velocity is still v_*=ṽ as the slit is stationary. Meanwhile we also imagine that the ball leaves from below the fictional slit at time t_* with velocity v_*=-ṽ (with an abuse of notation), then it collides at the next moving slit at time t̅ with outgoing velocity v̅.

Let us denote I_* = I(t_*,v_*), etc. We will need the following constants

    κ_Ii= 1/2m_+(ṁ_+ - m_+l̇_-/l_-)
       κ_Ii'= 1/2m_+l̇_-/l_-
       κ_Ii”= 1/8m_+l̈_-(1-1/3l_-^2)
       κ_Ii”'= 1/4m_+^2(l̈_- +1/6l_-m̈_+)
       κ_Ii””= 1/8m_+^3l̈_-
       κ_Ii””'= 1/8m_+^2m̈_+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:



Assume that (τ_1,ℐ_1) ∈ R_1^+ with ℐ_1 > V_* and that 

    l_2^- {ℒ_*ℐ_1(θ_2^* - θ_1^*) -τ_1 }_2 ≳ 2-f(t^*_2+)-f(t^*_2-).

Then the Poincaré map P_UL I^12:R_1^+ → R_2^- is given by 

    (ρ̅_2,𝒥̅_2)=G_UL I^12(τ_1,ℐ_1) + H_UL I^12(τ_1,ℐ_1) + 𝒪(ℐ_1^-2)
 where

    G_UL I^12(τ_1,ℐ_1)= (    l_2^-/m_2^+{ℒ_*ℐ_1(θ_2^* - θ_1^*) -τ_1 }_2 + m_2^+ - l_2^- -1/m_2^+,
       -m_2^+/l_2^-ℐ_1 + κ_I2(ρ̅_2-1) - κ_I2' )

and

    H_UL I^12(τ_1,ℐ_1)=(0 , κ_I2”/ℐ_1 +κ_I2”'ρ̅_2-1/I_1 +κ_I2””(ρ̅_2-1)^2/ℐ_1 -κ_I2””'(ρ̅_2-1)^3/ℐ_1).

Similarly, assume that (τ_2,ℐ_2) ∈ R_2^+ with ℐ_2 > V_* and that 

    l_1^- {ℒ_*ℐ_2(2+θ_1^* - θ_2^*) -τ_2 }_2 ≳ 2-f(t^*_1+)-f(t^*_1-).
 
Then the Poincaré map P_UL I^21:R_2^+ → R_1^- is given by 

    (ρ̅_1,𝒥̅_1)=G_UL I^21(τ_2,ℐ_2) + H_UL I^21(τ_2,ℐ_2) + 𝒪(ℐ_2^-2)
 where

    G_UL I^21(τ_2,ℐ_2)= (    l_1^-/m_1^+{ℒ_*ℐ_2(2+θ_1^* - θ_2^*) -τ_2 }_2 + m_1^+ - l_1^- -1/m_1^+,
       -m_1^+/l_1^-ℐ_2 + κ_I1(ρ̅_1-1) - κ_I1' )

and

    H_UL I^21(τ_2,ℐ_2)=(0 , κ_I1”/ℐ_2 +κ_I1”'ρ̅_1-1/ℐ_2 +κ_I1””(ρ̅_1-1)^2/ℐ_2 -κ_I1””'(ρ̅_1-1)^3/ℐ_2).




We present the proof of P_UL I^12. 
The formula for P_UL 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_*=ṽ. Then l_* = 1-f_* = ṽ(t_*-t̃)-l(t̃) and we have 

    I_* = L_*/2l_*ṽ,  τ_* = 0.

From the prop43 formula, 

    τ_* = l_-/l_*Ĩ(θ̃-θ_2^*) + 1 + l_-/l_* + 𝒪(v^-2)


    I_* = l_*/l_-Ĩ + L_*l_*^2/2l_-l̇_- -L_*^2l_*^3l̈_-/8Ĩ + L_*^2l_*^2l_*l̈_-/24Ĩ + 𝒪(v^-2).

Now we imagine that the ball leaves from below the fictional stationary wall at time t_* with outgoing velocity v_*=-ṽ (with an abuse of notation). Then m_*= -m(t̅) - v(t̅-t_*) and we have 

    J_* = -M_*/2m_*ṽ,  ρ_*=0.

From the prop44 formula, 

    ρ̅ = 1+m_*/m_+ + 𝒪(v^-2)


    J̅ = m_+/m_*J_* + M_*/2m_+ṁ_+(ρ̅-1) + M_*^2/8J_*m_+^2m_*m̈_+(ρ̅-1)^2 - M_*^2/24J_*m_+^2m_*m̈_+ + 𝒪(v^-2).

We observe that 

    l_*    = v(t_*-t̃)-l(t̃)
       = v(t_*-t̃)- l_- - l̇_-(t̃-t_*) + 𝒪(v^-2)
       = -(ṽ+l̇_-)(t_*-t̃)- l_- + 𝒪(v^-2)
       =-l_-Ĩ(θ̃-θ_2^*) - l_- + 𝒪(v^-2)

and that 

    m_*    = -m(t̅) - v(t̅-t_*)
       = -m_+ - ṁ_+(t̅-t_*) + (v̅+2ṁ_+)(t̅-t_*) + 𝒪(v^-2)
       = -m_+ + (v̅+ṁ_+)(t̅-t_*) + 𝒪(v^-2)
       = -m_+ +m_+ρ̅ + 𝒪(v^-2)

Since m_* = l_* -1, 

    ρ̅ = -l_-/m_+Ĩ(θ̃-θ_2^*) + m_+ - l_- -1/m_+ + 𝒪(v^-2).

Finally the relation between I_* and J_*, together with <Ref>, produces the formula for J̅.





If v(t_* -t) < f(t_*+) - f(t), then the ball enters the lower chamber immediately after it leaves the previous slit (c.f. <Ref> 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

    κ_ II i”= 1/4m_+^2l̈_-
       κ_ II i”'= 1/8m_+^2(l_-m̈_+ - m_+l̈_-)
       κ_ II i””= 1/24m_+l_-(l_-^2l̈_- - l_-m_+m̈_+ - 3l̈_-)

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




Assume that (τ_1,ℐ_1) ∈ R_1^+ with ℐ_1 > V_* and that 

    f(t^*_2+)-f(t^*_2-) ≳ l_2^- {ℒ_*ℐ_1(θ_2^* - θ_1^*) -τ_1 }_2.
 
Then the Poincaré map P_UL II^12:R_1^+ → R_2^- is given by 

    (ρ̅_2,𝒥̅_2)=G_UL II^12(τ_1,ℐ_1) + H_UL II^12(τ_1,ℐ_1) + 𝒪(ℐ_1^-2)
 where

    G_UL II^12(τ_1,ℐ_1)=(    l_2^-/m_2^+{ℒ_*ℐ_1(θ_2^* - θ_1^*) -τ_1 }_2 + m_2^+ - l_2^- +1/m_2^+,
       -m_2^+/l_2^-ℐ_1 + κ_ I 2(ρ̅_2-1) + κ_ I 2' )

and

    H_UL II^12(τ_1,ℐ_1)=(0 , -κ_ II 2”ρ̅_2-1/ℐ_1 -κ_ II 2”'(ρ̅_2-1)^2/ℐ_1 -κ_ II 2””/ℐ_1).

Similarly, assume that (τ_2,ℐ_2) ∈ R_2^+ with ℐ_2 > V_* and that 

    f(t^*_1+)-f(t^*_1-) ≳ l_1^- {ℒ_*ℐ_2(2+θ_1^* - θ_2^*) -τ_2 }_2.
 
Then the Poincaré map P_UL II^21:R_2^+ → R_1^- is given by 

    (ρ̅_1,𝒥̅_1)=G_UL II^21(τ_2,ℐ_2) + H_UL II^21(τ_2,ℐ_2) + 𝒪(ℐ_2^-2)
 where

    G_UL II^21(τ_2,ℐ_2)=(    l_1^-/m_1^+{ℒ_*ℐ_2(2+θ_1^* - θ_2^*) -τ_2 }_2 + m_1^+ - l_1^- +1/m_1^+,
       -m_1^+/l_1^-ℐ_2 + κ_ I 1(ρ̅_1-1) + κ_ I 1' )

and

    H_UL II^21(τ_2,ℐ_2)=(0 , -κ_ II 1”ρ̅_1-1/ℐ_2 -κ_ II 1”'(ρ̅_1-1)^2/ℐ_2 -κ_ II 1””/ℐ_2).




Again we only prove the formula for P_UL II^12.

We have from <Ref> that

    v(t̅-t̃)    = f(t̅) -f(t̃)
    ṽ( (t̅-t_*) - (t̃-t_*) )    = -m(t̅) + l(t̃) -1
     m_+ - (v̅+ṁ_+)(t̅-t_*)    = (ṽ+l̇_-)(t̃-t_*) + l_- -1 + 𝒪(v^-2)
     m_+ - m_+ ρ̅   = l_-Ĩ(θ̃-θ_2^*) +l_- -1  + 𝒪(v^-2)
    ρ̅   = -l_-/m_+Ĩ(θ̃-θ_2^*) + m_+ -l_- +1/m_+ + 𝒪(v^-2)

The computation is similar to that in the proof of Proposition <ref>. So we just list the key steps.

We observe that 

    2ℳ_*^-1J̅   = m̅v̅ + m̅ṁ̅̇+m̅^2m̈̅̈/3v̅
       =m_+v̅+m_+ṁ_+ +m_+ṁ_+ρ̅ + m_+m̈_+(t̅-t_*) + m_+^2m̈_+/3v̅
            + m̈_+/2(t̅-t_*)^2v̅ + 𝒪(v^-2)

and that

    2ℒ_*^-1Ĩ   = l̃ṽ + l̃l̇̃̇+l̃^2l̈̃̈/3ṽ
       =l_- ṽ + l_- l̇_- + l_-l̇_-Ĩ(θ̃-θ_2^*) + l_-l̈_-(t̃-t_*) + l_-^2l̈_-/3ṽ
            + l̈_-/2(t̃-t_*)^2ṽ + 𝒪(v^-2).

We also note from <Ref> that 

    ṽ=-v̅-2ṁ_+ -2m̈_+(t̅-t_*) + 𝒪(v^-2)

and that 

    t̅-t̃   = f(t̅)-f(t̃)/ṽ = -m(t̅) + l(t̃) -1/ṽ
       =-m_+ + l_- -1/ṽ + (-ṁ_+ +l̇_-)t̅-t_*/ṽ - l̇_-t̅-t̃/ṽ + 𝒪(v^-3)
       =-m_+ + l_- -1/ṽ + (-ṁ_+ +l̇_-)t̅-t_*/ṽ + l̇_-m_+ - l_- +1/ṽ^2 + 𝒪(v^-3).

Therefore 

    2ℒ_*^-1Ĩ   = -l_-v̅-2l_-ṁ_+ +m_+l̇_- + m_+l̇_-ρ̅ +l̇_-
            + (m_+l̈_- -2l_-m̈_+ + l̈_-)(t̅-t_*)- l̈_-/2(t̅-t_*)^2v̅
             -l̈_-/2v̅((m_+ +1)^2 - l_-^2/3) +𝒪(v^-2)

hence 

    2l_-/ℳ_*J̅ + 2m_+/ℒ_*Ĩ   = m_+l̇_- + m_+(l_-ṁ_+ -m_+l̇_-)(ρ̅-1) -1/2ℒ_*l_-m_+^2l̈_-ρ̅-1/I
            -1/4ℒ_*l_-m_+^2(l_-m̈_+ - m_+l̈_-)(ρ̅-1)^2/I
            -1/12Iℒ_*l_-m_+(l_-^2l̈_- - l_-m_+m̈_+ -3l̈_-) +𝒪(v^-2)

which produces the desired formula together with <Ref>.











  §.§.§ The Lower-Upper Chamber Case

Finally we suppose that (t̃,ṽ) ∈R̃ 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 (t̅,v̅)=F(t̃,ṽ).



If -v(t_* -t) > f(t)+f(t_*-), then the ball collides
with the floor and enters the upper chamber (c.f. <Ref> on the left).

The imaginary stationary wall trick also applies in this case, which produces a desired formula with the following constants

    χ_Ii= 1/2l_+(l̇_+ - l_+ṁ_-/m_-)
       χ_Ii'= 1/2l_+ṁ_-/m_-
       χ_Ii”= 1/8l_+m̈_-(1-1/3m_-^2)
       χ_Ii”'= 1/4l_+^2(1/6m_-l̈_+ -m̈_- )
       χ_Ii””= 1/8l_+^3m̈_-
       χ_Ii””'= 1/8l_+^2l̈_+m_-

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



Assume that (ρ_1,𝒥_1) ∈ R_1^- with 𝒥_1 > V_* and that -m_2^-{ℳ_*𝒥_1(ζ_2^* - ζ_1^*) -ρ_1 }_2 ≳ f(t^*_2-)+f(t^*_2+). Then the Poincaré map 
P_LU I^12: R_1^- → R_2^+ is given by 
    (τ̅_2,ℐ̅_2)=G_LU I^12(ρ_1,𝒥_1) + H_LU I^12(ρ_1,𝒥_1) + 𝒪(𝒥_1^-2)
 
where 

    G_LU I^12(ρ_1,𝒥_1)=(    m_2^-/l_2^+{ℳ_*𝒥_1(ζ_2^* - ζ_1^*) -ρ_1 }_2 + l_2^+ - m_2^- +1/l_2^+,
       -l_2^+/m_2^-𝒥_1 + χ_I2(τ̅_2-1) + χ_I2' )

and

    H_LU I^12(ρ_1,𝒥_1)=(0 , χ_I2”/𝒥_1 +χ_I2”'τ̅_2-1/𝒥_1 +χ_I2””(τ̅_2-1)^2/𝒥_1 -χ_I2””'(τ̅_2-1)^3/𝒥_1).

Similarly, assume that (ρ_2,𝒥_2) ∈ R_2^- with 𝒥_2 > V_* and that 

    -m_1^-{ℳ_*𝒥_2(2+ζ_1^* - ζ_2^*) -ρ_1 }_2 ≳ f(t^*_1-)+f(t^*_1+).
 
Then the Poincaré map P_LU I^21: R_2^- → R_1^+ is given by 

    (τ̅_1,ℐ̅_1)=G_LU I^21(ρ_2,𝒥_2) + H_LU I^21(ρ_2,𝒥_2) + 𝒪(𝒥_2^-2)
 
where 

    G_LU I^21(ρ_2,𝒥_2)=(    m_1^-/l_1^+{ℳ_*𝒥_2(2+ζ_1^* - ζ_2^*) -ρ_2 }_2 + l_1^+ - m_1^- +1/l_1^+,
       l_1^+/m_1^-𝒥_2 + χ_I1(τ̅_1-1) + χ_I1' )

and

    H_LU I^21(ρ_2,𝒥_2)=(0 , χ_I1”/𝒥_2 +χ_I1”'τ̅_1-1/𝒥_2 +χ_I1””(τ̅_1-1)^2/𝒥_2 -χ_I1””'(τ̅_1-1)^3/𝒥_2).




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. <Ref> on the right).

The computation as we have performed for Proposition <ref> can be reproduced here to present the formula in this case, the proof of which we ergo omit. We will need the following constants

    χ_ II i”= 1/4l_+^2m̈_-
       χ_ II i”'= 1/8l_+^2(m_-l̈_+ - l_+m̈_-)
       χ_ II i””= 1/24l_+m_-(m_-^2m̈_- - m_-l_+l̈_+ - 3m̈_-)

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



Assume that (ρ_1,𝒥_1) ∈ R_1^- with 𝒥_1 > V_* and that f(t^*_2-)-f(t^*_2+) ≳ -m_2^-{ℳ_*𝒥_1(ζ_2^* - ζ_1^*) -ρ_1 }_2. Then the Poincaré map 
P_LU II^12: R_1^- → R_2^+ is given by 

    (τ̅_2,ℐ̅_2)=G_LU II^12(ρ_1,𝒥_1) + H_LU II^12(ρ_1,𝒥_1) + 𝒪(𝒥_1^-2)
 
where

    G_LU II^12(ρ_1,𝒥_1)=(    m_2^-/l_2^+{ℳ_*𝒥_1(ζ_2^* - ζ_1^*) -ρ_1 }_2 + l_2^+ - m_2^- -1/l_2^+,
       -l_2^+/m_2^-𝒥_1 + χ_I2(τ̅_2-1) - χ_I2' )

and

    H_LU II^12(ρ_1,𝒥_1)=(0 , χ_ II 2”τ̅_2-1/𝒥_1 -χ_ II 2”'(τ̅_2-1)^2/𝒥_1 -χ_ II 2””/𝒥_1).

Similarly, assume that (ρ_2,𝒥_2) ∈ R_2^- with 𝒥_2 > V_* and that 

    f(t^*_1-)-f(t^*_1+) ≳ -m_1^-{ℳ_*𝒥_2(2+ζ_1^* - ζ_2^*) -ρ_2 }_2.
 
Then the Poincaré map P_LU II^21: R_2^- → R_1^+ is given by 

    (τ̅_1,ℐ̅_1)=G_LU II^21(ρ_2,𝒥_2) + H_LU II^21(ρ_2,𝒥_2) + 𝒪(𝒥_2^-2)
 where

    G_LU II^21(ρ_2,𝒥_2)=(    m_1^-/l_1^+{𝒥_2(2+ζ_1^* - ζ_2^*) -ρ_2 }_2 + l_1^+ - m_1^- -1/l_1^+,
       -l_1^+/m_1^-𝒥_2 + χ_ I 1(τ̅_1-1) - χ_ I 1' )

and

    H_LU II^21(ρ_2,𝒥_2)=(0 , χ_ II 1”τ̅_1-1/𝒥_2 -χ_ II 1”'(τ̅_1-1)^2/𝒥_2 -χ_ II 1””/𝒥_2).














§ TRAPPING REGIONS


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



We choose V_* ≫ 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 <ref> predicts the change of energy after one period in the lower chamber to be

    𝒥̅̅̅ = m_1^+/m_1^-m_2^+/m_2^-𝒥 + 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 m_1^+/m_1^-m_2^+/m_2^->1 in the 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 <ref> and the relative positions of the slits, its energy decreases at an exponential rate l_1^+/l_1^-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 

    𝒰_V={(t_0,v_0): V < v_0 < V+1,  lim sup v_n<V_*}

We claim that (𝒰_V)=0. Otherwise we note that 
𝒰_V ⊆{|v_n|<V+1, ∀ n>0}, which is bounded and invariant.
Hence, the Poincaré recurrence theorem implies that almost every point in 𝒰_V would return infinitely often to energy level |v_n| > V, which is impossible as it would contradict the definition of
𝒰_V. Our claim implies that almost all points in the energy shell 
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 <ref> guarantees an exponential energy gain at rate l_1^+/l_1^-l_2^+/l_2^->1 in the upper chamber once the ball gets trapped. The rest of the analysis is similar to Case (i).



 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 f̃_L(t)=f̃_R(t)=acos 4π t +0.5 for some small a>0. We take x_0=0, λ=0.5, so t_1^*=0.5, t_2^*=1.5. We consider a 4-periodic orbit 𝒫 starting at t_0=0.25, v_0=2+4a. 
Then

    (t_1, v_1)=(0.75,2+4a),    
    (t_2, v_2)=(1.25,2+4a),    
    (t_3, v_3)=(1.75,2+4a),    .

 
 We can slightly modify f̃_L near t_1^*, t_2^* in such a way that 
 
    f_L(0.5)>f_R(0.5),    f_L(1.5)<f_R(1.5)
 
 so that the upper chamber is trapping and that the periodic orbit 𝒫 does not see this modification. Observe that 𝒫 is elliptic for all small a as the trace of the collision map F along 𝒫 is tr (dF_𝒫) = 2-8aπ^2/1+2a∈ (0,2) for 0<a<1/4π^2-2. Now the matrix dF_𝒫 is conjugate to a rotation by 2πα with cos 2πα=1-4aπ^2/1+2a. We can easily choose a such that the rotation angle α is Diophantine, then Herman's Last Geometric Theorem guarantees the stability of the elliptic orbit 𝒫, i.e. there exists an elliptic island of bounded trajectories around 𝒫 (c.f. <cit.>).


Although the assumptions of Theorem <ref> 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 

    lim sup_t→+∞ |v(t)| =∞  and  lim inf_t→+∞ |v(t)| <∞.



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



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 <ref> 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.










§ 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∘ G_UU^12 and the hyperbolicity assumption |Tr|>2 indicates that G_U is hyperbolic.




 §.§ Almost Sure Escape for the Limiting Map

We first restrict ourselves to the linear parts G_UU's of the dynamics in Proposition <ref>, 
which approximates P_UU's with an error of order 𝒪(I^-1) when the velocity is large v>V_*.

We note that 

    (τ̅_2,I̅_2)   =G_UU^12(τ_1,I_1)
       =( -l_2^-/l_2^+{ℒ_*ℐ_1(θ_2^* - θ_1^*) -τ_1 }_2 + 1+ l_2^-/l_2^+, l_2^+/l_2^-ℐ_1 + Δ_2 (τ̅_2 -1) )

if 1-l_2^+/l_2^- < {ℒ_*ℐ_1(θ_2^* - θ_1^*) -τ_1 }_2 < 1+l_2^+/l_2^-. The boundary lines 

    {ℒ_*ℐ_1(θ_2^* - θ_1^*) -τ_1 }_2=1-l_2^+/l_2^-,   {ℒ_*ℐ_1(θ_2^* - θ_1^*) -τ_1 }_2=1+l_2^+/l_2^-

cut out from R_1^+ a sequence of boxes 

    A_n={1-l_2^+/l_2^- +2n < ℒ_*ℐ_1(θ_2^* - θ_1^*) -τ_1 < 1+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 

    (τ̅_1,I̅_1)   =G_UU^21(τ_2,I_2)
       =( -l_1^-/l_1^+{ℒ_*ℐ_2(2+θ_1^* - θ_2^*) -τ_2 }_2 + 1+ l_1^-/l_1^+, l_1^+/l_1^-ℐ_2 + Δ_1 (τ̅_1 -1) )

if 1-l_1^+/l_1^- < {ℒ_*ℐ_2(2+θ_1^* - θ_2^*) -τ_2 }_2 < 1+l_1^+/l_1^- and that the boundary lines 

    {ℒ_*ℐ_2(2+θ_1^* - θ_2^*) -τ_2 }_2=1-l_1^+/l_1^-,   {ℒ_*ℐ_2(2+θ_1^* - θ_2^*) -τ_2 }_2=1+l_1^+/l_1^-

cut out from R_2^+ another sequence of boxes 

    B_n={1-l_1^+/l_1^-+2n<ℒ_*ℐ_2(2+θ_1^* - θ_2^*) -τ_2<1+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∘ 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· DG_UU^12 where

    DG_UU^12 =
       [                                       l_2^-/l_2^+                   -l_2^-/l_2^+ℒ_* (θ_2^* - θ_1^*);                                   Δ_2 l_2^-/l_2^+ -Δ_2 l_2^-/l_2^+ℒ_* (θ_2^* - θ_1^*) + l_2^+/l_2^- ],



    DG_UU^21 =
       [                                         l_1^-/l_1^+                   -l_1^-/l_1^+ℒ_* (2+θ_1^* - θ_2^*);                                     Δ_1 l_1^-/l_1^+ -Δ_1 l_1^-/l_1^+ℒ_* (2+θ_1^* - θ_2^*) + l_1^+/l_1^- ]


Since (DG_U)=1 and |Tr(DG_U)|>2, it has unstable eigenvalue Λ_u with unstable eigenvector e_u, and stable eigenvalue Λ_s with stable eigenvector e_s.


We observe that each box A is foliated by unstable lines.

We say that an unstable line γ 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:


  Let γ be a good unstable line in some box A. Then the proportion of points on γ which remain in the upper chamber after one period is at most 
 
    D=1+2l_1^+/l_1^-/2+l_1^+/l_1^-<1.



We first note that G_UU^12(γ) remains a complete piece in R_2 as γ lies in A and that G_UU^12 maps the boundaries of A into two vertical lines 

    G_UU^12({ℒ_*ℐ_1(θ_2^* - θ_1^*) -τ_1 }_2=1-l_2^+/l_2^-) ⊆{τ_2=2},
 

    G_UU^12({ℒ_*ℐ_1(θ_2^* - θ_1^*) -τ_1 }_2=1+l_2^+/l_2^-) ⊆{τ_2=0}.


G_UU^12(γ) has to stretch across at least two B-boxes if γ has at least two pieces remaining in the upper chamber after one period.


Suppose that G_UU^12(γ) 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(γ) ends on the boundaries of the top and bottom boxes as shown in Figure <ref>. However the height of a B-box is equal to 2l_1^+/l_1^- (c.f. (<ref>)) while the height of the fundamental domain is equal to 2 (c.f. Figure <ref>). This implies that in the optimal situation 2(1-l_1^+/l_1^-)(N-1)/2N+2l_1^+/l_1^- of the points on G_UU^12(γ) land in the lower chamber after jumping from left to right at t_1^*. This proportion is larger than (1-l_1^+/l_1^-)(2-1)/2+l_1^+/l_1^- as it is an increasing function in N and N≥ 2. Then the largest portion which remains in the upper chamber is given by 
D=1+2l_1^+/l_1^-/2+l_1^+/l_1^-. 
Recall that the relative positions of two slits at t_1^* implies that 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 γ is an unstable line in some box A. For x ∈γ, we denote as r_n(x) the distance from x_n to the nearest boundary of the component γ_n containing x_n. Employing the argument in Section 5 of <cit.> we obtain the following Growth Lemma.


  There exists a constant C^* s.t. for any small ϵ>0 and any n∈ℕ 
  
    _γ{x∈γ: r_n(x)<ϵ}≤ C^* ϵ



 Let k_n(δ) denote the max number of the pieces that an unstable line of length less than δ can be cut into. We define k_n=lim_δ→ 0 k_n(δ). We claim that k_n ≤ 8n. Indeed since the singularities of G_U^n are lines and there are at most 8n possibilities for slopes. Consequently, there exists δ_0 so small that k_n(δ) ≤ 16n for any δ<δ_0. We choose n_0 such that 32n_0/Λ_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 δ_0 and let r̅_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 r̅_n(x) ≤ r_n(x) and it suffices to prove the statement for r̅_n.

 First we observe that 

    _γ{r̅_0(x)<ϵ}≤2L/δ_0ϵ .

 r̅_n+1(x) is less than ϵ if x_n+1 either passes a real or artificial singularity, where L is the unstable height of γ. The former is controlled by 
 2k_1 (δ_0) _γ{ r_n<ϵ/Λ_u} while the latter by 2k_1 (δ_0)L/δ_0ϵ. Therefore 
 
    _γ{r̅_n+1<ϵ}≤32/Λ_u_γ{ r_n<ϵ} + 32L/δ_0ϵ.

 Thus by induction we conclude that 
 
    _γ{r̅_n<ϵ}≤(32/Λ_u)^n _γ{r̅_0(x)<ϵ} + 32L/δ_0ϵ(1+⋯ +(32/Λ_u)^n-1).

 Since 32/Λ_u<1, (<ref>)
gives the desired growth control with 

    C^*=(32/Λ_u)2L/δ_0+32L/δ_0(1-32/Λ_u).
   


Finally we show that under the linear approximation map G_U almost every point will eventually escape to the trapping region:

   In each box A, for any ϵ>0, there exists N=N(ϵ) such that all but an ϵ-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.



Fix ϵ>0. Choose k,l such that 

    D^k L< 0.5 ϵ  and   (kl+1) C^*+L^2/Λ_u^l/2 < 0.5 ϵ,
 and take N=kl+1.

We suppose that under the linear map G_U a point x on a unstable line γ 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 <ref> 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 

    _γ{ x∈γ : {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, ⋯, x_n+l land on bad lines γ_n,⋯,γ_n+l for some n<N and we denote as ℬ_n the set of all such x∈γ that lands badly during n to n+l periods. We subdivide ℬ_n into ℬ_n,L and ℬ_n,S, where ℬ_n,L collects points with |γ_n| ≥Λ_u^-l/2 and ℬ_n,S collects points with |γ_n| < Λ_u^-l/2.

By Lemma <ref>, |ℬ_n,S| ≤ C^* Λ_u^-l/2. On the other hand, it follows from uniform hyperbolicity that 

    _γ_n{x_n returns badly for next l periods} = |γ_n+l|/Λ_u^l≤L/Λ_u^l.
 
Hence 

    |ℬ_n,L| ≤L/Λ_u^l∑_|γ_n| ≥Λ_u^-l/2 |G_U^-nγ_n| ≤L/Λ_u^l |γ| ≤L^2/Λ_u^l.


Combining the  estimates on ℬ_n,L 
and ℬ_n,S, 
we have 

    |ℬ_n| ≤C^*/Λ_u^l/2 + 
    L^2/Λ_u^l≤C^*+L^2/Λ_u^l/2.

Consequently the set ℬ of points on γ which make l consecutive bad landings is controlled in size by 

    |ℬ| ≤ (kl+1) C^*+L^2/Λ_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 ϵ. 






 §.§ Quick Escape for the Actual Map.

By Proposition <ref>, the fundamental domains of P_UU are 𝒪(ℐ^-1)-deformation of the boxes A,B and P_UU=G_UU+𝒪(ℐ^-1).

Now we prove Theorem <ref>.



Fix ϵ>0 and a box A with large energy ℐ̃_0 (to be specified later). By Proposition <ref> 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 ϵ, i.e. we take N=kl+1 where k,l are integers such that 

    k>log (0.25ϵ/L)/log D  and  kl+1/Λ_u^l/2 < 0.25ϵ/C^*+L^2.

We shall show that the statement of Theorem <ref> holds with some large ℐ̃_0=ℐ̃_0(ϵ) and

    T=2N.


Let A_n^δ and B_n^δ denote the points in A_n and B_n which are closer than
δ to the boundary, 

    𝒜^δ=⋃_n A_n^δ,   𝒜^δ=⋃_n (A_n-∖ A_n^δ),


    ℬ^δ=⋃_n B_n^δ,   ℬ^δ=⋃_n (B_n-∖ B_n^δ).

 Choose δ<0.5 ϵ so that the set of points in the box A which visit either 𝒜^δ or 
 ℬ^δ during the first N iterations is less than 0.5 ϵ.
 
By Proposition <ref>, 
there is a constant C_1 such that if 
P_UU^12(x)∈ℬ^C_1 /ℐ and 
P_UU^21(P_UU^12 x)∈𝒜^C_1 /ℐ 
then the orbit of x stays in the upper chamber for the next period and 

    | P_U(x)-G_U(x)|≤C_1/ℐ  where  
    P_U=P_UU^21∘ P_UU^12.


Accordingly there is a constant C_2 such that if for some n≤ N

    P_UU^12 P_U^k (x)∈ℬ^C_2Λ_u^N/ℐ^*,   
    P_U^k+1 (x)∈𝒜^C_2 Λ_u^N/ℐ^*

and ℐ_k≥ℐ^*
for k < n then the real orbit of x stays in upper chamber for the first n iterations
and

    | P_U^n(x)-G_U^n (x)|≤C_2 Λ_u^N/ℐ^*.

 
Next, set C_3=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 (<ref>)
and ℐ_0≥ C_3^N ℐ^* then (<ref>) holds.


Now choose ℐ^* so that 

    C_2 Λ_u^N/ℐ^*<δ



Now we consider the orbits where ℐ̅≤ℐ_0≤ℐ̅+1
for some ℐ̅≥ C_3^N 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 𝒜^δ for the first N iterations;

(iii) The real orbit of x stays in 𝒜^δ until it hits 
𝒜^δ∪ 
(P_U^12)^-1ℬ^δ
at some period n<N.

Proposition <ref> and our choice of δ and ℐ^* imply that the set of orbits where either (ii) or (iii) happens has measure smaller than ϵ.

This completes the proof of Theorem <ref>.














§ 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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