## View abstract

### Session S39 - Differential Equations and Geometric Structures

Monday, July 19, 16:00 ~ 16:50 UTC-3

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

### Svetlana Katok

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 Teichmüller space, taking all values (flexibility'') between zero and a maximum that is achieved on the surface which 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.