An integrable deformation of an ellipse of small eccentricity is an ellipse. (English) Zbl 1379.37104

A strictly convex domain \(\Omega \subset \mathbb{R}^2\) is \(C^r\) if its boundary is a \(C^r\)-smooth curve. This very interesting paper concerns the billiard problem inside a \(C^r\) domain \(\Omega\), called {billiard table}.
A (possibly not connected) curve \(\Gamma \subset \Omega \) is called {caustic} if any billiard orbit having one segment tangent to \(\Gamma \) has all its segments tangent to \(\Gamma \). Then the billiard \(\Omega \) is called {locally integrable} if the union of all caustics has nonempty interior; \(\Omega \) is called {integrable} if the union of all {smooth convex} caustics has nonempty interior.
It is well known that an ellipse billiard is integrable since its caustics are cofocal ellipses and hyperbolas. A long standing open question is whether or not there exist integrable billiards that are different from ellipses. The Birkhoff conjecture states that integrability implies the fact that \(\partial \Omega \) is ellipse.
The present work proves a version of this conjecture for tables bounded by small perturbations of ellipses of small eccentricity. A main remark is that {infinitesimally} rationally integrable deformations of a circle are tangent to a five-parametric family of ellipses.


37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37D50 Hyperbolic systems with singularities (billiards, etc.) (MSC2010)
37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
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