Elliptic Islands on the Elliptical Stadium

We investigate the existence of elliptic islands for a special family of periodic orbits of a two-parameter family of maps corresponding to the billiard problem on the elliptical stadium. The hyperbolic or elliptical character of these orbits is also investigated. Depending on the parameters, we obtain upper bounds of ellipticity for this special family as a lower bound for chaos. On a different region of the parameter space, we can prove that there is no upper bound for the existence of elliptic islands. The main results we use are Birkhoff Normal Form and Moser's Twist Theorem.


Introduction
The elliptical stadium is a plane region bounded by a curve Γ, constructed by joining two half-ellipses, with major axes a > 1 and minor axes b = 1, by two straight segments of equal length 2h (see fig. 1). 2h a 1 Figure 1: The elliptical stadium.
The billiard on the elliptical stadium consists in the study of the free motion of a point particle inside the stadium, being reflected elastically at the impacts with Γ. Since the motion is free inside Γ, it is determined either by two consecutive points of reflection at Γ or by the point of reflection and the direction of motion immediately after each collision.
For fixed a and h, let s ∈ [0, L) be the arc length parameter for Γ and the direction of motion be given by the angle β with the normal to the boundary at the impact point. The billiard defines an invertible map T a,h from the annulus A = [0, L) × (−π/2, π/2) into itself, preserving the measure dµ = cos β dβ ds. 0 AMS subject classification: 37E40, 70K42 Since Γ is globally C 1 but not C 2 , T a,h is a homeomorphism (see, for instance, [6]) and if (s 0 , β 0 ) and (s 1 , β 1 ) = T a,h (s 0 , β 0 ) ∈ A such that Γ is analytic in some neighborhoods of s 0 and s 1 , then clearly T a,h is analytic in some neighborhoods of (s 0 , β 0 ) and (s 1 , β 1 ).
For each (a, h), (A, µ, T a,h ) defines a discrete dynamical system, whose dynamics depends on the values of a and h. For instance, when h = 0 we have an ellipse and the billiard is integrable.
When h = 0, two main features appear. If a < √ 2, Donnay [3] proved that the billiard on the elliptical stadium is chaotic (in the sense of non-vanishing Lyapunov exponents) when h is sufficiently large. Lower bounds for h for this behaviour were found by Markarian and ourselves in [5] and by Canale, Markarian and ourselves in [2]. In the present work we show that the lower bound found in [2] is optimal, in the sense that bellow it we can assure the existence of elliptic islands of positive measure.
In [1] Bunimovich conjectured the existence of a stable periodic orbit, with island of positive measure, for billiards such as the elliptical stadium with a > √ 2 and h = 0. In this work we make some progress in this direction, proving that there is no upper bound on h for the existence of elliptic islands if a > √ 2. So, there is no way to destroy the elliptic islands by just increasing the distance between the half-ellipses.
To prove the existence of elliptic islands we extend the results about a special family of periodic orbits, called pantographic, studied in [2] and find regions on the parameter plane where at least one of its members is elliptic and stable, so, with islands of positive measure in phase space.

Pantographic orbits: existence and ellipticity
In this section we define the special family of periodic orbits and investigate the existence and ellipticity of its members. This family has already been investigated in [2] for a < √ 2. Here we extend that work for all a > 1. We will skip most of the proofs which can be found in the work cited above.
Given a, h and a positive integer n, an (n, a, h)-pantographic orbit, denoted by P an(n, a, h), is a symmetric (4 + 2n)-periodic orbit, with exactly 2 impacts at each half-ellipse, joined by a vertical path, and crossing any vertical line only twice. One example can be seen in figure 2. Let the right half-ellipse of the stadium be parametrized by (x, y) = (a cos λ + h, sin λ) and P be the point marked on figure 2. Using the obvious symmetries (see figure 3), the parameter λ of P must satisfy: tan 2β = a tan λ a 2 tan 2 λ − 1 = h + a cos λ n + sin λ and tan β = cos λ a sin λ where β > 0 is, as defined above, the angle of the trajectory from P , with the normal to the boundary. The following proposition gives the region of existence of those pantographic orbits in the parameter plane Proposition 1 • P an(0, a, h) and P an(1, a, h) exist for every a > 1 and h > 0.
For n = 0 and n = 1 this results implies the existence of the corresponding P an(n, a, h) for every a > 1 and h > 0.
However, for n ≥ 2 one must also ask that tan 2β ≥ a cos λ 1 + sin λ in order to guarantee that the next impact point from P is on the straight part of the boundary. This is equivalent to which is always true if 1 < a ≤ 2.
If a > 2 we rewrite (3) as 0 ≤ t = tan λ ≤ 1 √ a(a−2) . Since ∂t ∂h (n, a, h) < 0 (which is easily verified from (2)), and 1 a < 1 √ a(a−2) , there exists a unique h such that t(n, a, h) = and if h > h, . From (2), h = (n − 1) a(a − 2). ♣ For each fixed n, we denote by U n the open region in the parameter plane where P an(n, a, h) exists, according to Proposition 1. For (a, h) ∈ U n , let s be the arc length corresponding to the point P of P an(n, a, h) and β the angle with the normal of the trajectory at this point as before. Then T 4+2n a,h (s, β) = (s, β) and the ellipticity of this orbit is determined by the eigenvalues of DT 4+2n a,h | (s,β) .
As shown in [2], we can write DT 4+2n and where l 1 is the length of the trajectory between two impacts with the same half-ellipse, l 2 is the length of the trajectory between two impacts with the different half-ellipses and K is the curvature of the ellipse at s.
h) < 1 then P an(n, a, h) is elliptic (meaning that the eigenvalues of DT 4+2n | (s,β) are unitary with non zero imaginary part).
The following lemma summarizes some properties of ∆ n (a, h) and its technical proof has been postponed to the appendix.
The function ∆ n (a, h) has the following properties: For each 0 < c ≤ 1 + 2n, let α c n be the unique solution of L n (a) = c. We have that 1 < α c n < √ 2 and if c < d then α d n < α c n .
It follows from the lemma that every level curve ∆ n = c is given by a graph h c n : (α c n , +∞) → IR such that ∆ n (a, h c n (a)) = c and ∆ n (a, h) < c if h < h c n (a) and ∆ n (a, h) > c if h > h c n (a).
The characterization of the region of ellipticity is then given by:  figure 4).
For each n and (a, h) in the region of ellipticity of P an(n, a, h), let µ and µ be the eigenvalues of DT 4+2n a,h | (s,β) . P an(n, a, h) has a resonance of order k if µ k = 1, i.e., µ = e i 2jπ k , j = 1, ..., k − 1. It is easy to show that this corresponds, for k > 1, to Clearly 0 < c jk < 1 and we have the curves of resonance given by the graphs of h

Existence of elliptic islands
In order to establish if the elliptic periodic orbits described in the previous section have invariant curves surrounding them, we will invoke the two classical results: Birkhoff Normal Form: Let f be an area preserving map in C l (l ≥ 4 ) with a fixed point at the origin, with eigenvalues µ and µ, |µ| = 1. If for some integer q in 4 ≤ q ≤ l + 1 one has µ k = 1 for k = 1, 2, ..., q then there exists a real analytic transformation taking f into the normal form where τ (ζζ) = τ 1 |ζ| 2 + ... + τ s |ζ| 2s , with s = q 2 − 1, is a real polynomial in |ζ| 2 and g vanishes with its derivatives up to order q − 1 at ζ = ζ = 0.
Theorem ( Moser, [8], p.56) If the polynomial τ (|ζ| 2 ) does not vanishes identically, ζ = 0 is a stable fixed point (which means that there are invariant curves surrounding it, and so an elliptic island of positive measure).
For each fixed period n 0 , we will then investigate the resonances of T 4+2n0 a,h and the zeros of the coefficients of its Birkoff normal form, near the Pantographic orbit.
Let us fix a period n 0 and a major axis a 0 > α 1 n0 = (2 + 2n 0 )/(2 + n 0 ). According to proposition 4, P an(n 0 , a 0 , h) is elliptic with no resonances up to order q if h is in the finite union of open disjoint adjacent intervals denoted ∪I q j .
Let λ(h), s(h), β(h) be respectively the parameter, the arc length and the angle with the normal for the point P of P an(n 0 , a 0 , h).
Lemma 5 For any q ≥ 4, λ(h), s(h) and β(h) are analytic functions of h on each I q j .
Proof: The point P will belong to P an(n 0 , a 0 , h) if t = tan λ satisfies equation (2): are analytical functions of t and A(t) = 0, since 1 is analytic. As this equation has a unique solution for each h, the inverse t = t(h) exists and is then locally analytic for every h ∈ I q j .
The functions λ(h) = arctan t(h) and the corresponding arc length of the ellipse s(h) = s(λ(h)) are then analytic. The same is true for β(h) = β(λ(h)). ♣ For each fixed h, let f be the translation of T 4+2n0 a0,h by (s(h), β(h)). The map f is clearly area preserving and analytic in (s, β) on a neighbourhood of the origin. The eigenvalues of Df (0,0) are the same as those of DT 4+2n0 a0,h (s(h), β(h)). If h ∈ I q j , f can be written in the Birkhoff normal form (4). If one of the Birkkoff coefficients is not zero, f has an elliptic island surrounding (0, 0) and, by translation, there is an elliptic island surrounding P an(n 0 , a 0 , h). Let (s, β) and (s ′ , β ′ ) be two consecutive impacts of a trajectory with the two different half-ellipses (with l ≥ 0 impacts with the straight parts between them), or two consecutive impacts of a trajectory with the same half-ellipse (with l = 0 ). Then DT 4+2n0 a0,h (s, β), for (s, β) near to (s(h), β(h)) is a finite product of matrices of the form (see, for instance, [6]) where K stands for the curvature of the ellipse at the impact and L is the total length of the trajectory between the two impacts with the half-ellipses.
Since cos β = 0 for β near β(h), all the entries of the matrix above , as well as their derivatives up to any order in s and β, are analytic functions of h. Using lemma 5 we conclude that all the coefficients of J q−1 f (0,0) are analytic in h.
If follows that τ m (h) is analytic in h, leading immediately to the next corollary. ♣

Corollary 7
On each I q j and for 1 ≤ m ≤ q 2 − 1, the set {h / τ m (h) = 0} is either the entire I q j or a discrete set.
In order to prove the existence of islands we use the natural recurrence on the order of the ressonaces.
We begin by analysing the zeros of τ 1 on the non resonat intervals I 4 j . If τ 1 = 0 only on a discrete subset of each I 4 j , P an(n 0 , a 0 , h) has elliptic island except for a discrete set of values of h (which can be smaller than the union of the discrete subsets of zeros of τ 1 and the values of resonance up to order 4, since on the discrete subsets a non resonnat value of higher order may have a non zero Birkhoff coefficient).
If τ 1 is identically zero on one of those intervals, we proceed to the next step, aplying the same analysis to the zeros of τ 2 .
We continue the recurrence and it will end up in a finite number of steps if for some order of resonance the last Birkhoff coefficient does not vanish identically on a whole non resonant interval. Otherwise, all the Birkhoff coefficients will vanish on at least one open interval, bounded by resonant values of h. In this last case, since ∂∆n ∂h > 0, the rotation number ρ(h) of P an(n 0 , a 0 , h) is not constant and there exists h 0 such that ρ(h 0 ) is diophantine. As f is analytic, it is conjugate to a rotation and there will be invariant curves [4].
We conclude that: Theorem 1 Given n and a > α 1 n , there are at least countably many values of h in ∪I 4 j such that P an(n, a, h) has an elliptic island.
Remark: As Moeckel proved in [7], in a generic one-parameter family of area preserving maps with elliptic fixed points, the first Birkhoff coefficient τ 1 varies from −∞ to +∞ as the rotation number varies from 0 to 1/3 or from 2/3 to 1. So, we do not expect τ 1 (h) to be always different from zero, neither to vanish identically. Furthermore, generically, at a zero of τ 1 , a higher Birkhoff coefficient will not vanish. So, although we can handle the case τ s (h 0 ) = 0, ∀s, we do not expect it to happen in our case.
4 Bounds for the existence of islands 4.1 The case a < √ 2 As shown in proposition 3 and in [2], if (a, h) is in the region of ellipticity of P an(n, a, h) then a ≥ α 1 n = 2n+2 n+2 . So, given a ∈ (1, √ 2), there is only a finite number of periods 4 + 2n such that P an(n, a, h) can be elliptic. More precisely, n ≤ 2(a 2 −1) 2−a 2 .
Let H(a) be the maximum of h 1 n (a) for those periods. As proved in [2], H(a) is a lower bound for chaos. By theorem 1, it is also an upper bound for the existence of elliptic islands for the Pantographic family.

The case a > √ 2
On the other hand, if a > √ 2, P an(n, a, h) can be elliptic for any period n. Moreover, for each n and q, ∪I q j ⊂ (h 0 n (a), h 1 n (a)), with h 0 n (a) = na √ a 2 − 2. We also have that (h 0 n (a), h 1 n (a)) ∩ (h 0 n (a), h 0 n+1 (a)) is a non empty open interval. So we can find h, na √ a 2 − 2 < h < (n + 1)a √ a 2 − 2 such that P an(n, a, h) has an elliptic island. This proves the following Theorem 2 Given a > √ 2 there is no upper bound on h for the existence of elliptic islands on the elliptical stadium billiard.
However, as can be seen in figure 5, for values of a further away from √ 2, the strips of ellipticity are disjoint. In these gaps all pantographic orbits are hyperbolic, having, thus, no islands. Nevertheless, in our simulations other islands appear, obviously corresponding to different periodic orbits. In figure 6 we exemplify this fact for a = 2 and h = 2, a value located in the gap between the strips of ellipticity for n = 0 and n = 1 (see figure 5). We show three non-pantographic orbits and their islands and the whole phase space, where we can see many other islands surrounded by what seems to be a chaotic sea. As far as our results indicate and our simulations show, this should be the typical picture for the phase space when a > √ 2. 5 Appendix

Some properties of t(n, a, h)
We called U n the open region in the parameter plane where P an(n, a, h) exists: for n ≥ 2.
The following lemma gives some useful information about t(n, a, h), the solution of (1) or (2).
For n ≥ 2, we remark first that if k > l then t(k, a, h) > t(l, a, h). So, for 1 < a ≤ 2 lim h→0 + t(n, a, h) = +∞. For a > 2, the limit as h → (n−1) a(a − 2) is the unique solution of equation (1) for h = (n−1) a(a − 2) which is t = 1 √ a(a−2) . ♣ Remark: When h → 0 + , the elliptical stadium becomes an ellipse. t → +∞ means that the pantographic orbit goes to the elliptic periodic orbit which corresponds to the minor axis of the ellipse. Let us call pantographic-like orbits in the elliptical billiard the periodic trajectories that have vertical segments both at left and right extremes. As can be seen in [5], the 4-periodic pantographic-like orbit exists if a > √ 2 and the 6-periodic if a > 2, they are parabolic and their position is given, respectively, by t = . They are the calculated limits of the 4 and 6-periodic pantographic orbits of the elliptical stadium.

Proof of the lemma 2
For a fixed n, let (a, h) ∈ U n and λ n (a, h) be the solution of (1), β be the angle, with the normal, of the outgoing trajectory at P = (a cos λ n + h, sin λ n ) and s the corresponding arc length.
This shows that l2 K cos β is the product of two positive decreasing functions of t and so ∂ ∂t l2 K cos β < 0.
Lemma 2 For every n ≥ 0 the function ∆ n (a, h) has the following properties: