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

Abrams, Adam; Katok, Svetlana; Ugarcovici, Ilie

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

Given a closed, orientable surface of constant negative curvature and genus $g\ge 2$, 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 $(8g-4)$-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.

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

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