Bialy-Mironov type rigidity for centrally symmetric symplectic billiards

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 $δ\subset \mathcal{P}$ of rotation number $1/4$ (winding once around $\partial D$) and consisting only of $4$-periodic orbits. If one of the parts between $δ$ 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 $δ$, and the result is obtained by tracing back the problem directly to the totally integrable case.

Figures (3)

• Figure 1: The symplectic billiard map.
• Figure 2: The same colored regions have equal area.
• Figure 3: The angle $\alpha$ and the support function $p(\alpha)$.

Theorems & Definitions (12)

• Theorem 1.1
• Corollary 1.2
• Corollary 1.3
• Proposition 3.1
• proof
• Proposition 3.2
• Lemma 4.1
• proof
• Lemma 4.2
• proof