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arXiv · 1909.07032

Flexibility of measure-theoretic entropy of boundary maps associated to Fuchsian groups

Adam Abrams·Svetlana Katok·Ilie Ugarcovici
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Abstract

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 Teichm\"uller 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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Adam Abrams, Svetlana Katok, Ilie Ugarcovici. 2019-09-16. Flexibility of measure-theoretic entropy of boundary maps associated to Fuchsian groups. https://arxiv.org/abs/1909.07032

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