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

Part of: Dynamical systems with hyperbolic behavior Low-dimensional dynamical systems Measure-theoretic ergodic theory Other groups of matrices

Published online by Cambridge University Press:  14 April 2021

ADAM ABRAMS ,
SVETLANA KATOK  and
ILIE UGARCOVICI
ADAM ABRAMS*
Affiliation:
Institute of Mathematics, Polish Academy of Sciences, Warsaw00656, Poland
SVETLANA KATOK
Affiliation:
Department of Mathematics, The Pennsylvania State University, University Park, PA16802, USA (e-mail: [email protected])
ILIE UGARCOVICI
Affiliation:
Department of Mathematical Sciences, DePaul University, Chicago, IL60614, USA (e-mail: [email protected])
*
e-mail: [email protected]
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Abstract

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.

Keywords

Fuchsian groupsboundary mapsentropyTeichmüller space

MSC classification

Primary: 37D40: Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) 37E10: Maps of the circle
Secondary: 28D20: Entropy and other invariants 20H10: Fuchsian groups and their generalizations
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 2: Anatole Katok Memorial Issue Part 1: Special Issue of Ergodic Theory and Dynamical Systems , February 2022 , pp. 389 - 401
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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Footnotes

In memory of Tolya

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