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

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 present and sketch the proofs of two strikingly different results: topological entropy is constant in this entire family (“rigidity”), while measure-theoretic entropy varies within Teichm"uller space, taking all values (“flexibility”) between zero and a maximum, which is achieved on the surface that admits a regular fundamental $(8g-4)$-gon. We obtain explicit formulas for both entropies. The rigidity proof uses conjugation to maps of constant slope, while the flexibility proof – valid only for certain multi-parameters – uses the realization of geodesic flow as a special flow over the natural extension of the boundary map.