On the local Birkhoff conjecture for convex billiards. (English) Zbl 1394.37093

This paper considers the classical Birkhoff conjecture that the boundary of a strictly convex integrable billiard table must be an ellipse (or, as a special case, a circle). The conjecture is still unresolved. The authors prove a complete local version that a small integrable perturbation of an ellipse must be an ellipse.
The main result is as follows: assume that \({\mathcal E}\) is an ellipse with eccentric \(e_0\), \(0\leq e_0<1\), and semi-focal distance \(c\). Provided that \(k\geq 39\) for every \(K>0\), there is an \(\varepsilon=\varepsilon(e_0,c,K)\) such that if \(\Omega\) is a rationally integrable \(C^k\)-smooth domain with a boundary \(\partial\Omega\) \(C^k\)-\(K\)-closed and \(C^1\)-\(\varepsilon\)-close to \({\mathcal E}\), then \(\Omega\) is an ellipse. Here it is assumed that \(\partial\Omega\) consists of \({\mathcal E}\) plus a \(C^k\)-perturbation \(\mu\) with \(\| \mu\|_{C^k}\leq K\) and \(\|\mu\|_{C^1}<\varepsilon\). (The latter describe \(C^k\)-\(K\)-close and \(C^1\)-\(\varepsilon\)-close.)
The result here parallels similar recent results of A. Avila et al. [Ann. Math. (2) 184, No. 2, 527–558 (2016; Zbl 1379.37104)] and G. Huang et al. [Duke Math. J. 167, No. 1, 175–209 (2018; Zbl 1417.37138)]. A critical idea in this paper enables the authors to move beyond the prior results in the almost-circular case. This was to consider analytic extensions of the action-angle coordinates of elliptic billiards (i.e., the boundary parametrizations that are induced by the integrable caustics) and to carefully evaluate their singularities. The authors express such functions in terms of elliptic integrals and Jacobi elliptic functions.


37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
70H06 Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics
37E40 Dynamical aspects of twist maps
33E05 Elliptic functions and integrals
37D50 Hyperbolic systems with singularities (billiards, etc.) (MSC2010)
Full Text: DOI arXiv


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