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

Given a closed, orientable, compact surface S of constant negative curvature and genus, 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-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-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.