Rigidity of topological entropy of boundary maps associated to Fuchsian groups

TL;DR

This paper proves that the topological entropy of a family of boundary maps related to Fuchsian groups remains constant, depending only on the genus of the surface, and provides an explicit formula for it.

Contribution

It establishes a rigidity result showing the topological entropy is invariant within the family and across the Teichmüller space, with an explicit entropy formula.

Findings

Topological entropy is constant within the family of boundary maps.

Entropy depends only on the genus of the surface.

Explicit formula for the topological entropy is provided.

Abstract

Given a closed, orientable surface of constant negative curvature and genus $g \ge 2$, we study a family of generalized Bowen-Series boundary maps and prove the following rigidity result: in this family the topological entropy is constant and depends only on the genus of the surface. We give an explicit formula for this entropy and show that the value of the topological entropy also stays constant in the Teichm\"uller space of the surface. The proofs use conjugation to maps of constant slope.