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 the paper we show that a version of this conjecture is true for tables bounded by small perturbations of ellipses of small eccentricity.

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...

The Birkhoff conjecture says that the boundary of a strictly convex integrable billiard table is necessarily an ellipse. In this article, we consider a stronger notion of integrability, namely integrability close to the boundary, and prove a local version of this conjecture: a small perturbation of an ellipse of small eccentricity which preserves integrability near the boundary, is itself an ellipse. This extends the result in [1], where integrability was assumed on a larger set. In particular, it shows that (local) integrability near the boundary implies global integrability. One of the crucial ideas in the proof consists in analyzing Taylor expansion of the corresponding action-angle coordinates with respect to the eccentricity parameter, deriving and studying higher order conditions for the preservation of integrable rational caustics.3 The same remark applies to rotation numbers 3 2q , for q not divisible by 3.

In this survey, we provide a concise introduction to convex billiards and describe some recent results, obtained by the authors and collaborators, on the classification of integrable billiards, namely the so-called Birkhoff conjecture.This article is part of the theme issue 'Finite dimensional integrable systems: new trends and methods'.