中心对称辛弹球的 Bialy–Mironov 型刚性
Bialy-Mironov type rigidity for centrally symmetric symplectic billiards
AI总结：
本文证明中心对称强凸区域上的辛弹球若存在旋转数 1/4 的 4-周期轨道不变曲线，且一侧相空间由闭不变曲线叶化，则边界必为椭圆。
摘要：
本文旨在建立中心对称辛弹球的 Bialy–Mironov 型刚性。对于具有边界 ∂D 的中心对称 C^2 强凸区域 D，假设辛弹球映射存在一条（简单）连续不变曲线 δ⊂P，其旋转数为 1/4（绕 ∂D 缠绕一周），并且该曲线完全由 4-周期轨道组成。如果 δ 与相空间各边界之间的两个区域之一，完全由连续的、闭的（非零伦的）不变曲线叶化，则 ∂D 是椭圆。与 Birkhoff 弹球的区别主要有两点：可以假设在曲线 δ 所划分出的相空间其中一个区域内存在叶状结构；并且该结果是通过将问题直接追溯到完全可积情形而得到的。
英文摘要：
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.
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