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

TL;DR

This paper derives an explicit formula for the measure-theoretic entropy of boundary maps associated with Fuchsian groups, showing its variability across Teichmüller space and comparing it to topological entropy.

Contribution

It provides a new explicit entropy formula depending only on polygon perimeter and genus, and demonstrates the entropy's flexibility within Teichmüller space.

Findings

Entropy varies continuously between 0 and a maximum

Maximum entropy achieved by regular polygon surfaces

Smooth invariant measure is not the maximal entropy measure

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.