Effective rigidity away from the boundary for centrally symmetric billiards

IF 0.8 3区数学 Q2 MATHEMATICS

Ergodic Theory and Dynamical Systems Pub Date : 2023-09-28 DOI:10.1017/etds.2023.70

MISHA BIALY

引用次数: 0

Abstract

Abstract In this paper, we study centrally symmetric Birkhoff billiard tables. We introduce a closed invariant set $\mathcal {M}_{\mathcal {B}}$ consisting of locally maximizing orbits of the billiard map lying inside the region $\mathcal {B}$ bounded by two invariant curves of $4$ -periodic orbits. We give an effective bound from above on the measure of this invariant set in terms of the isoperimetric defect of the curve. The equality case occurs if and only if the curve is a circle.

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中心对称台球远离边界的有效刚性

摘要本文研究中心对称Birkhoff台球桌。我们引入了一个闭不变集$\mathcal {M}_{\mathcal {B}}$，它由位于$\mathcal {B}$区域内的台球图的局部最大化轨道组成，该区域由$ $4$ $周期轨道的两条不变曲线所包围。根据曲线的等周缺陷，给出了该不变集的测度的有效界。当且仅当曲线为圆时，等式成立。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Ergodic Theory and Dynamical Systems 数学-数学

CiteScore

1.70

自引率

11.10%

发文量

113

审稿时长

6-12 weeks

期刊介绍： Ergodic Theory and Dynamical Systems focuses on a rich variety of research areas which, although diverse, employ as common themes global dynamical methods. The journal provides a focus for this important and flourishing area of mathematics and brings together many major contributions in the field. The journal acts as a forum for central problems of dynamical systems and of interactions of dynamical systems with areas such as differential geometry, number theory, operator algebras, celestial and statistical mechanics, and biology.