## 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.

### MSC:

 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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