Given a closed, orientable, compact surface S of constant negative curvature and genus $g \geq 2$ , 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 a measure of maximal entropy.

中文翻译：

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与 Fuchsian 群相关的边界图的测量理论熵的灵活性

给定一个封闭的、可定向的、紧凑的表面小号恒定负曲率和属$g \geq 2$，我们研究了 Bowen-Series 边界图关于其平滑不变测度的测度理论熵。我们得到了一个只依赖于周长的熵的显式公式$(8g-4)$面的边基本多边形小号及其属。使用它，我们分析了 Teichmüller 空间中的熵如何变化小号并证明以下灵活性结果：测度理论熵取介于 0 和最大值之间的所有值，该最大值在允许规则的表面上实现$(8g-4)$边基本多边形。我们还将测度理论熵与这些映射的拓扑熵进行比较，并表明平滑不变测度不是最大熵的测度。