与富克斯群相关的边界映射的测度熵的柔性
Flexibility of measure-theoretic entropy of boundary maps associated to Fuchsian groups
摘要：
给定一个具有常负曲率和亏格$g \ge 2$的紧致定向闭曲面$S$，研究了Bowen系列边界映射关于其光滑不变测度的测度熵.我们得到了一个显式的熵公式，它只依赖于周长的$(8g-4)$边的基本多边形的表面$S$和它的亏格。利用这一点，我们分析了如何在Teichmüller空间的$S$的熵的变化，并证明了以下灵活性的结果：测量理论的熵采取所有的值之间的$0$和一个最大值，是在表面上，允许一个规则的$(8g-4)$边的基本多边形。我们还比较了这些映射的测度熵和拓扑熵，证明了光滑不变测度不是最大熵的测度。
英文摘要：
Given a closed, oriented, compact surface $S$ of constant negative curvature and genus $g \ge 2$, we study the measure-theoretic entropy of the Bowen-Series boundary map with respect to its smooth invariant measure. We obtain an explicit formula for the entropy that only depends on the perimeter of the $(8g-4)$-sided fundamental polygon of the surface $S$ and its genus. Using this, we analyze how the entropy changes in the Teichmüller space of $S$ and prove the following flexibility result: the measure-theoretic entropy takes all values between $0$ and a maximum that is achieved on the surface that admits a regular $(8g-4)$-sided fundamental polygon. We also compare the measure-theoretic entropy to the topological entropy of these maps and show that the smooth invariant measure is not the measure of maximal entropy.
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