Rigidity and flexibility of entropies of boundary maps associated to Fuchsian groups

Adam Abrams; Svetlana Katok; Ilie Ugarcovici

doi:10.5802/mrr.10

Adam Abrams ¹; Svetlana Katok ²; Ilie Ugarcovici ³

¹ Faculty of Pure and Applied Mathematics, Wrocław University of Science and Technology, Wrocław 50370, Poland
² Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA
³ Department of Mathematical Sciences, DePaul University, Chicago, IL 60614, USA

Mathematics Research Reports, Volume 3 (2022), pp. 1-19.

Abstract

Given a closed, orientable surface of constant negative curvature and genus $g \geq 2$ , we study the topological entropy and measure-theoretic entropy (with respect to a smooth invariant measure) of generalized Bowen–Series boundary maps. Each such map is defined for a particular fundamental polygon for the surface and a particular multi-parameter.

We survey two strikingly different recent results by the authors: topological entropy is constant in this entire family (“rigidity”), while measure-theoretic entropy varies within Teichmüller space, taking all values (“flexibility”) between zero and a maximum. This maximum is achieved on the surface that admits a regular fundamental $(8 g - 4)$ -gon. The rigidity proof uses conjugation to maps of constant slope, while the flexibility proof—valid for a large set of multi-parameters—uses the realization of geodesic flow as a special flow over the natural extension of the boundary map. We obtain explicit formulas for both entropies. We also present some new details pertaining to specific multi-parameters and particular polygons.

Received: 2021-06-18
Revised: 2022-04-14
Published online: 2022-05-23

DOI: 10.5802/mrr.10

Classification: 37D40, 37E10
Keywords: Fuchsian groups, boundary maps, entropy, topological entropy, flexibility, rigidity

Author's affiliations:

Adam Abrams ¹; Svetlana Katok ²; Ilie Ugarcovici ³

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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Adam Abrams; Svetlana Katok; Ilie Ugarcovici. Rigidity and flexibility of entropies of boundary maps associated to Fuchsian groups. Mathematics Research Reports, Volume 3 (2022), pp. 1-19. doi : 10.5802/mrr.10. https://mrr.centre-mersenne.org/articles/10.5802/mrr.10/

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