Perfect Retroreflectors and Billiard Dynamics

We construct semi-infinite billiard domains which reverse the direction of most incoming particles. We prove that almost all particles will leave the open billiard domain after a finite number of reflections. Moreover, with high probability the exit velocity is exactly opposite to the entrance velocity, and the particle's exit point is arbitrarily close to its initial position. The method is based on asymptotic analysis of statistics of entrance times to a small interval for irrational circle rotations. The rescaled entrance times have a limiting distribution in a limit when the number of iterates tends to infinity and the length of the interval vanishes. The proof of the main results follows from the study of related limiting distributions and their regularity properties.


Introduction
The present paper is motivated by the problem of constructing open billiard domains with exact velocity reversal (EVR), which means that the velocity of every incoming particle is reversed when the particle eventually leaves the domain. This problem arises in the construction of perfect retroreflectors-optical devices that exactly reverse the direction of an incident beam of light and preserve the original image. A well-known example of a perfect retroreflector is the Eaton lens [3], [12] which is a spherically symmetric lens that, unlike our model, also reverses the original image. A second application lies in the search for domains that maximize the pressure of a flow of particles [10]: for a particle of mass m > 0, which moves towards a wall with velocityv, the impulse transmitted to the wall at the moment of reflection is equal to 2m|v n |, wherev n is the normal component ofv. It is maximized whenv =v n , i.e. when the direction of the particle is reversed.
We construct a family of domains D , for which EVR holds up to a set of initial condition whose measure tends to zero in the limit ε → 0. The domain D is the semi-infinite tube [0, ∞)×[0, 1] with vertical barriers of height /2 at the points (n, 0) and (n, 1), n ∈ N as illustrated in Fig. 1. Inside the domain the particle moves with the constant speed and elastic reflections from the boundary. Since the kinetic energy of the particle is preserved, we can assume that the speed of the particle is equal to one. The motion of the particle is determined by the point y in ∈ [0, 1], where it enters the tube and the initial velocity v in = (cos(πϕ), sin(πϕ)) at this point. The measure P on the initial conditions (y in , ϕ) considered below is a Borel probability measure absolutely continuous with respect to the Lebesgue measure on Ω = [0, 1] × [−1/2, 1/2]. Theorem 1. For every ε ∈ (0, 1) there exists a set Ω(ε) ⊂ Ω of full Lebesgue measure, such that for every (y in , ϕ) ∈ Ω(ε), the particle eventually leaves the tube.
The position and the velocity with which it leaves the tube are denoted by (y out , v out ). By Theorem 1, for every ε ∈ (0, 1) the functions y out = y out (y in , v in ) and v out = v out (y in , v in ) are defined P-almost everywhere.
Theorem 2. For any δ > 0, Theorem 2 follows from the results on the existence of certain limiting distributions for the exit statistics of the billiard particle as ε → 0. Below we formulate these results as Theorem 3 and Theorem 4. In the last section of the paper we show how they imply Theorem 2.
Let Q ε = Q ε (y in , v in ) be the number of reflections from the vertical walls before the particle leaves the tube. Let T ε = T ε (y in , v in ) be the time that particle spends inside the tube. By Theorem 1, both Q ε and T ε are finite P−a.e.
Consider also a bi-infinite tubular domain similar to the one described above. It consists of two horizontal lines at the unit distance from each other and a one-periodic configuration of vertical walls of height ε/2. Let x be the horizontal coordinate and assume that the particle starts at x = 0. Let ξ 0 ε = 0 and ξ k ε ∈ Z be x-coordinate of the particle at the moment of k'th reflection from a vertical wall. Since the tube is now bi-infinite, {ξ k ε } is a discrete time process on Z, defined for any k ∈ N. We also define a continuous version of this process: {ξ ε (t)} is the projection of the trajectory of a billiard particle in the bi-infinite tube to the x-axis normalized to have constant speed 1/ε.
(2) There exists a limiting probability distribution function G : The second part of Theorem 3 says that for the limiting stochastic process {ξ k }, with probability one there exists k ∈ N, such that ξ k < 0. Similar results are true for the continuous process {ξ ε (s)} as well: (1) The process {εξ ε (s)} converges in distribution w.r.t. P to a stochastic process ξ(s) as ε → 0.
(2) There exists a limiting probability distribution function H :

Reduction to Circle Rotations and Point-Wise Exits
We first reformulate the problem in terms of circle rotations. Let us identify [0, 1) with S 1 = R/Z. For α ∈ R, let R α : S 1 → S 1 be the circle rotation by angle α : Always assume that α ∈ R \ Q.
We define several sequences measuring the return times to the interval I ε , which will be used throughout the proofs. The hitting times m k ε = m k ε (x, α), k = 0, 1, 2, . . . are defined for x ∈ S 1 by: : R l α x ∈ I ε } The sequence n k ε = n k ε (x, α), k = 1, 2, . . . of relative return times to the interval I ε is defined for x ∈ S 1 by: We shall also use the sequence {ξ k ε } defined in the introduction as the sequence of the horizontal coordinates of points of the reflection from the vertical walls.
Note that if x = y in , and α = tan(πϕ), then n i ε (x) is the distance between horizontal coordinate of the place of the (i − 1)'st and the i'th reflections of the particle from vertical walls. Therefore, The probability measure P on the initial conditions (y in , ϕ in ) ∈ [0, 1] × [−1/2, 1/2] for the billiard particle induces a probability measure on the initial conditions (x, α) ∈ [0, 1) × [0, 1) T 2 for the circle rotation R α , which is absolutely continuous w.r.t. to the Lebesgue measure on T 2 and which will be also denoted by P.
Proof. The proof of Proposition will follow from a combination of results of [1] and [2].
Recall Property P introduced by Boshernitzan in [1]: has an infinite number of solutions (n 1 , n 2 , . . . n l ).
Let m n (T α,ε ) be the length of the smallest interval of continuity ofT n α,ε .
Definition 2. An interval exchange mapT α,ε has property P, if for some δ > 0 the set is essential.
By Theorem 5.3 of [2], property P implies weak-mixing for an interval exchange of three intervals with combinatorics (3, 2, 1) (and more generally, for any combinatorics of a so-called W −type, see [2]).
The next two statements are well-known. We include their proofs to keep the exposition self-contained.
Proposition 8. Let T be an ergodic transformation on (X, µ), µ(X) = 1, Proof. Since T is ergodic, the set of points x for which S n (f, x) is bounded from below has measure either equal to zero or one. In the first case, Proposition is proved, so assume that it has measure one. Then the function It is enough to show that hdµ = 0. If g(x) ∈ L 1 (X, µ), then hdµ = 0 by the definition of h(x) above. If not, then by Birkhoff ergodic theorem, for µ-a.e. x ∈ X, where the integral can be equal to infinity. We write is finite almost everywhere, we can choose a set Y ⊂ X, such that µ(Y ) > 0 and for every y ∈ Y, |g(y)| < M for some constant M. Then by ergodicity of T, there exists a subsequence n k , such that T n k x ∈ Y, and therefore, by Birkhoff ergodic theorem for µ-almost all x ∈ X we have Proof of Theorem 1. For any ε > 0 choose an α ∈ Λ(ε), so that the map on a positive Lebesgue measure set. In either case, the ergodicity ofT 2 α,ε , implies that for Lebesgue a.e. x, there exists

Limiting distributions
We now prove theorems 3 and 4.

Notations and the formulation of the main limiting
the number of times the particle hits vertical walls during the time ε −1 T.
. . , n}. Let χ I denote the characteristic function of the interval I ⊂ R and ψ T (x, y) = χ (0,1] (x)χ [−T /2,T /2] (y) be the characteristic function of a corresponding rectangle. Then Let ASL(2, R) = SL(2, R) R 2 be the semidirect product group with multiplication law The action of an element (M, v) of this group on R 2 is defined by Each affine lattice of covolume one in R 2 can then be represented as Z 2 g for some g ∈ ASL(2, R), and the space of affine lattices is represented by X = ASL(2, Z)\ASL(2, R), where ASL(2, Z) = SL(2, Z) Z 2 . Denote by ν the Haar probability measure on X.
Theorem 9. As ε → 0, the limit of (3.3) exists and is equal to which is a C 1 function R n ≥0 → [0, 1]. We define the associated limiting probability density φ (n) (t 1 , . . . , t n ) by The convergence for the random variable Q ε (x, α) also follows from Theorem 9.
Indeed, for any k ≥ 1 let χ A k be the characteristic function of the set ∆ k = {(y 1 , . . . y k ) ∈ R k : y 1 > 0, . . . , y k−1 > 0, y k < 0}. Then for every ε > 0 we have and by Theorem 9 and the Helly-Bray Theorem ( [5], p.183), there exists the limit Notice that the representation (3.7) implies that    Let us enumerate points of g which lie in R ∞ according to their horizontal coordinates: if the coordinates of the k'th lattice point of g in R ∞ are (x k , y k ) = (x k (g), y k (g)) (k = 1, 2, . . .), then x k < x k+1 for any k = 1, 2, . . . Notice, that ν-almost every lattice g has infinitely many points in R ∞ .
Therefore, in order to prove (3.8), it is enough to show that for ν-almost every affine lattice g ∈ X, there exists an even k > 0, such that We will now show that the sequence y k (g) is an orbit of a certain map of an interval into itself, reduce (3.9) to a Birkhoff sum over this map and treat it in the way as in Section 2.
First, we describe the map. Consider set I ⊂ R 2 of vertical segments of unit length centered at every lattice point of g. We identify each segment in I with the base I = [−1/2, 1/2] of the tube R ∞ by parallel translation. Let π : I → I be the projection, which sends a point on some interval through a lattice point to the corresponding point in I.
Consider a unit speed flow in the positive horizontal direction on R 2 . Its first return map to I is a well-defined mapT =T (g) of I into itself. We define the corresponding invertible map T : I → I, so that π •T = T • π. It is easy to see, that the map T is an exchange of three intervals. For ν-almost every lattice g it has combinatorial type (3 2 1).
For every y ∈ I, we let ψ(y) to be the Euclidean distance betweenŷ ∈ π −1 (y) and its image underT . Clearly, this does not depend on the choice ofŷ ∈ π −1 (y).
Notice that the sequence {y k } of the vertical coordinates of the lattice points of g in R ∞ is related to the map T described above: for k ∈ N, y k = −T k−1 (−y 1 ) (see Figure 2). Also for k ∈ N, we have ψ(−y k ) = x k+1 − x k . Let −y 0 = T −1 (−y 1 ). Then the sum in (3.9) has the form (recall, k is even) Therefore similarly to Section 2, the alternating sum (3.9) is reduced to a Birkhoff sum for the function f (y) = ψ(−y) − ψ(−T (−y)) and the map T 2 .
Let the lengths of the interval exchange map T be equal to (λ 1 , λ 2 , 1 − λ 1 − λ 2 ). Denote the simplex of possible λ i 's by and the corresponding interval exchange map of combinatorial type (3, 2, 1) by T λ 1 ,λ 2 . The following theorem was first proved by Katok and Stepin in [4].
Similarly to the proof of Theorem 1, Theorem 11 and Proposition 8 imply that there exists a full Lebesgue measure subset Λ 1 ⊂ Λ, such that for every (λ 1 , λ 2 ) ∈ Λ 1 , there exists a full Lebesgue measure subset I = I (λ 1 , λ 2 ) ⊂ I, such that for every y ∈ I there exists k > 0, such that LetX ⊂ X be the set of lattices, for which the construction above gives an interval exchange transformation of combinatorial type (3 2 1). ThenX is open and ν(X) = 1. Notice that for any g ∈X, the map X : g → (λ 1 , λ 2 , y 0 ) is differentiable and its differential is surjective. Therefore, the preimage of any Lebesgue measure zero set under X has Haar measure zero in X. Therefore, the set of lattices g ∈ X, such that X (g) ∈ {(λ 1 , λ 2 , y 0 ) | (λ 1 , λ 2 ) ∈ Λ 1 , y 0 ∈ I (λ 1 , λ 2 )} has full Haar measure in X and so (3.8) is proved.
3.4. The proof of Theorem 9. By (3.3) it is enough to show that for any n ∈ N and any n-tuples (t 1 , . . . , t n ) ∈ R n >0 , k = (k 1 , . . . , k n ) ∈ Z n ≥0 there exists the limit and that G (n) (t 1 , . . . , t n ) is a C 1 -function of (t 1 , . . . , t n ). For n = 1 the convergence in (3.16) was first proved by Mazel and Sinai ([9]). It was later reproved and generalized by the third author ( [6], [7]) using different methods. The proof of the convergence in (3.16) follows the one in [6]. The proof of the regularity of the limiting function is similar to the one in [8].
We reduce the convergence in (3.16) to an equidistribution result for the geodesic flow on X.
Recall, that the action of the geodesic flow on X is given by right action of a one-parameter subgroup of X : The unstable horocycle of the flow Φ t on X is then parametrized by the subgroup H = {n − (x, α)} (x,α)∈T 2 : For g ∈ X let F T (g) be equal to the number of lattice points of Z 2 g in the rectangle R(T ). Then by (3.4) N ε (x, α, T ) = F T (n − (x, α)Φ t ) with t = −2 ln(ε).
Theorem 14. [6] For any bounded f : ASL(2, Z)\ASL(2, R) → R, such that the discontinuities of f are contained in a set of ν-measure zero and any Borel probability measure P, absolutely continuous with respect to Lebesgue measure on [0, 1) × [0, 1) , 0 otherwise Then D(g) satisfies the conditions of Theorem 14. The convergence in (3.16) now follows from theorem 14 applied to the function D(g).
We now prove C 1 regularity of the limiting function G (n) (t 1 , . . . , t n ). It is enough to consider the case when all t j are different. We also assume that all k j > 0. The case when some k j = 0 is similar.
Let X 1 = SL(2, Z)\SL(2, R) be the homogeneous space of lattices of covolume one and let ν 1 be the probability Haar measure on X 1 . For a given y ∈ R 2 let X(y) = {g ∈ X : y ∈ Z 2 g}, where the action of X on R 2 is given by the formula (3.5).
There is a natural identification of the sets X(y) and X 1 through Under this identification the probability Haar measure ν 1 on X 1 induces a probability Borel measure ν y on X(y). We will need the following two results.
Proposition 15. (Siegel's formula, [11]) Let f ∈ L 1 (R 2 ), then Proposition 16. ( [8]) Let E ⊂ X be any Borel set; then y → ν y (E ∩ X(y)) is a measurable function from R 2 to R. If U ⊂ R 2 is any Borel set such that E ⊂ ∪ y∈U X(y), then Futhermore, if ∀y 1 = y 2 ∈ U : X(y 1 ) ∩ X(y 2 ) ∩ E = ∅, then equality holds in (3.18) Notice that Propositions 15 and 16 imply that if there are two different indices 1 ≤ i, j ≤ n, such that Consider a single term in the expression above.
and let U = R(t j + h j ) \ R(t j ). Then by the proposition 16, Therefore, by proposition 15, For every fixed y ∈ [−1/2, 1/2] continuity of the expression under the integral sign with respect to (t 1 , . . . , t n ) again follows from Proposition 15. It is clearly uniform in y and therefore the integral is continuous with respect to (t 1 , . . . , t n ). Each term in (3.19) can be treated in a similiar way and this proves C 1 regularity of the function G (n) (t 1 , . . . , t n ) and finishes the proof of Theorem 9.
By (4.21) this is the case, when εQ ε < min{y in , 1 − y in } By the assumption, the probability measure P on the initial conditions (y in , α) is absolutely continuous with respect to the Lebesgue measure, therefore for any k ∈ N, P{Q ε = k, εk < min{y in , 1 − y in }} = P{Q ε = k} − P{Q ε = k, min{y in , 1 − y in } ≤ εk} → G(k) as ε → 0 Together with the tightness condition (3.11) this implies P{εQ ε < min{y in , 1 − y in }} → 1 as ε → 0 and so, P{v out = −v in } → 1 as ε → 0. Note that the existence of the limiting probability distribution for {Q ε } as ε → 0 also implies that for any δ > 0, P{|y out − y in | > δ} → 0 as ε → 0. Indeed, after each reflection from a vertical wall, the particle backtracks itself with an error at most ε, so at the moment of exit it backtracks the incoming trajectory with total error of at most εQ ε .
This finishes the proof of Theorem 2.