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The Birkhoff-Poritsky conjecture for centrally-symmetric billiard tables. (English) Zbl 1505.37067

The authors prove the Birkhoff-Poritsky conjecture for centrally-symmetric \(C^2\)-smooth convex planar billiards. G. D. Birkhoff [Acta Math. 50, 359–379 (1927; JFM 53.0733.03)] was the first to explore billiards in strictly convex planar domains. A conjecture that has been attributed to Birkhoff was then formalized by H. Poritsky [Ann. of Math. (2) 51 (1950), 446–470 (1950; Zbl 0037.26802)]. This conjecture asks if the only integrable convex billiards are ellipses.
The setting for the paper is as follows: \(\gamma \subset \mathbb{R}^2\) is a simple closed centrally-symmetric \(C^2\) curve of positive curvature with a fixed counterclockwise rotation, and \(\mathbf{A}\) is the phase cylinder of the billiard map \(T\) (the space of all oriented lines intersecting \( \gamma \)). The authors’ main result is the following. Suppose that the map \(T\) of \(\gamma \) has a continuous rotational invariant curve \(\alpha \subset \mathbf{A}\) (winding once around the cylinder and simple) of rotation number \(\frac{1}{4}\) consisting of 4-periodic orbits. If \(\mathcal{A} \subset \mathbf{A}\) is the domain between the curve \(\alpha\) and the boundary \(\delta = 0 \) of the phase cylinder (where \(\delta\) denotes the incoming angle of the line), then if \(\mathcal{A}\) is foliated by continuous rotational invariant curves, \(\gamma\) is an ellipse.
One of the main ingredients in the proof of this result is a non-standard generating function for convex billiards; another is an integral geometry approach for circular billiards. The new generating function leads to a twist map with respect to another vertical foliation of the phase cylinder \(\mathbf{A}\).
The authors note that the notion of integrability is important for approaching the problem. Their work is based on the rigidity of “total integrability”, which implies the existence of a foliation of the whole phase space by invariant tori.

MSC:

37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
37J39 Relations of finite-dimensional Hamiltonian and Lagrangian systems with topology, geometry and differential geometry (symplectic geometry, Poisson geometry, etc.)
37J51 Action-minimizing orbits and measures for finite-dimensional Hamiltonian and Lagrangian systems; variational principles; degree-theoretic methods
37C83 Dynamical systems with singularities (billiards, etc.)

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