TY - JOUR

T1 - Flexibility of measure-Theoretic entropy of boundary maps associated to Fuchsian groups

AU - Abrams, Adam

AU - Katok, Svetlana

AU - Ugarcovici, Ilie

N1 - Publisher Copyright:
© 2022 Cambridge University Press. All rights reserved.

PY - 2022/2/14

Y1 - 2022/2/14

N2 - 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.

AB - 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.

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U2 - 10.1017/etds.2021.14

DO - 10.1017/etds.2021.14

M3 - Article

AN - SCOPUS:85104421777

SN - 0143-3857

VL - 42

SP - 389

EP - 401

JO - Ergodic Theory and Dynamical Systems

JF - Ergodic Theory and Dynamical Systems

IS - 2

ER -