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.

中文翻译：

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中心对称台球远离边界的有效刚性

在本文中，我们研究中心对称伯克霍夫台球桌。我们引入一个闭不变集 $\mathcal {M}_{\mathcal {B}}$ 由位于该区域内的台球图的局部最大化轨道组成 $\mathcal {B}$ 由两条不变曲线界定 $4$ -周期性轨道。我们从上面根据曲线的等周缺陷给出了这个不变集的度量的有效界限。当且仅当曲线是圆时，才会发生相等情况。