On the local Birkhoff conjecture for convex billiards

Abstract

The classical Birkhoff conjecture claims that the boundary of a strictly convex integrable billiard table is necessarily an ellipse (or a circle as a special case). In this article we prove a complete local version of this conjecture: a small integrable perturbation of an ellipse must be an ellipse. This extends and completes the result in [3], where nearly circular domains were considered. One of the crucial ideas in the proof is to extend actionangle coordinates for elliptic billiards into complex domains (with respect to the angle), and to thoroughly analyze the nature of their complex singularities. As an application, we are able to prove some spectral rigidity results for elliptic domains. Dedicated to the memory of our thesis advisor John N. Mather: a great mathematician and a remarkable person arXiv:1612.09194v5 [math.DS] 1 Poritsky was Birkhoff's doctoral student and [41] was published several years after Birkhoff's death. 2 We are grateful to M. Bialy for pointing out this reference. 3 This regime of integrability is somehow diametrically opposed to ours, since we are interested in integrability near the boundary of the billiard domain. ON THE LOCAL BIRKHOFF CONJECTURE FOR CONVEX BILLIARDS 5Let us introduce an important notion for this paper.Definition 6. (i) We say Γ is an integrable rational caustic for the billiard map in Ω, if the corresponding (non-contractible) invariant curve Γ consists of periodic points; in particular, the corresponding rotation number is rational.(ii) If the billiard map inside Ω admits integrable rational caustics of rotation number 1/q for all q > 2, we say that Ω is rationally integrable.Remark 7. A simple sufficient condition for rational integrability is the following (see [3, Lemma 1]). Let C Ω denote the union of all smooth convex caustics of the billiard in Ω; if the interior of C Ω contains caustics of rotation number 1/q for any q > 2, then Ω is rationally integrable.Our main result is the following.Main Theorem (Local Birkhoff Conjecture). Let E 0 be an ellipse of eccentricity 0 ≤ e 0 < 1 and semi-focal distance c; let k ≥ 39. For every K > 0, there exists ε = ε(e 0 , c, K) such that the following holds: if Ω is a rationally integrable C k -smooth domain so that ∂Ω is C k -K-close and C 1 -ε-close to E 0 , then Ω is an ellipse.Remark 8. One could replace the smallness condition in the C 1 -norm with a smallness condition with respect to the C 0 -topology (this can be showed by using interpolation inequalities and the convexity of the domains) 4 .Remark 9. In [21] we prove a similar rigidity statement for a different type of rational integrability. Namely, we describe an algorithm to prove that for any given q 0 ≥ 3 there exists e 0 = e(q 0 ) > 0 such that every sufficiently smooth perturbation of E e , with 0 < e < e 0 , having integrable rational caustics of rotation numbers p/q, for all 0 < p/q < 1/q 0 , must be an ellipse. This algorithm is conditional on checking the invertibility of finitely many explicit matrices, which we prove in the cases q 0 = 3, 4, 5. Obse...

Robin Spectral Rigidity of the Ellipse

“…We see that there exist many notions of integrability, yet Birkhoff's conjecture remains open for all of them. However, as mentioned in Section 2, a local version was proven in [KS18].…”

“…A short proof of this can be found using elementary planar geometry in the appendix of [GM79a]. For elliptical caustics, we follow the notation in [KS18] and [DDCRR17] by setting λ 2 = Z ≥ 0. In the context of [KS18], integrable is taken to mean that the union of all convex caustics has a non-empty interior in R 2 .…”

“…To evaluate this quantity, we will use action angle coordinates for elliptical billiards, which we now describe, following the presentation in [KS18], [DDCRR17] and [CF88]. We begin by developing some basic elliptic function theory.…”

On the Finite Dimensionality of Integrable Deformations of Strictly Convex Integrable Billiard Tables

“…Though much attention it has attracted, this conjecture remains open, and only a few partial progresses were obtained. As far as our understanding of integrable billiards is concerned, the most important related results are 1) a theorem ( [2]) by Bialy which asserts that if the phase space of a billiard map is almost everywhere foliated by non-null homotopic invariant curves, then the corresponding billiard table is a disk; 2) a result ( [9]) by Innami, in which he showed that if a strictly convex billiard table admits a sequence of smooth convex caustics with rotation numbers converge to 1/2, then its boundary has to be an ellipse; 3) a result ( [5]) by Delshams and Ramírez-Ros in which they study entire perturbations of elliptic billiards and prove that any nontrivial symmetric perturbations of the elliptic billiard is not integrable (see also [4,13]); 4) and the more recent works ( [1,8,10]) by Kaloshin et al, justifying a perturbative version of the Birkhoff conjecture for billiard tables with boundary close to ellipses, assuming integrability near the boundary.…”

“…The domain enclosed by an ellipse is 2-rationally integrable. Due to [10], any smooth one parameter family of deformations of this domain, preserving the 2-integrability, are consisted of a family of domains with ellipses as their boundary, belonging to a 5-dimensional space. Theorem 1.3 can be viewed as a finite-dimensional reduction for integrable deformations.…”