Rigidity of topological entropy of boundary maps associated to Fuchsian groups

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.