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arXiv · 2402.19154

Bialy-Mironov type rigidity for centrally symmetric symplectic billiards

Luca Baracco·Olga Bernardi·Alessandra Nardi
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Abstract

The aim of the present paper is to establish a Bialy-Mironov type rigidity for centrally symmetric symplectic billiards. For a centrally symmetric $C^2$ strongly-convex domain $D$ with boundary $\partial D$, assume that the symplectic billiard map has a (simple) continuous invariant curve $\delta \subset \mathcal{P}$ of rotation number $1/4$ (winding once around $\partial D$) and consisting only of $4$-periodic orbits. If one of the parts between $\delta$ and each boundary of the phase-space is entirely foliated by continuous invariant closed (not null-homotopic) curves, then $\partial D$ is an ellipse. The differences with Birkhoff billiards are essentially two: it is possible to assume the existence of the foliation in one of the parts of the phase-space detected by the curve $\delta$, and the result is obtained by tracing back the problem directly to the totally integrable case.

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Luca Baracco, Olga Bernardi, Alessandra Nardi. 2024-02-29. Bialy-Mironov type rigidity for centrally symmetric symplectic billiards. https://arxiv.org/abs/2402.19154

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