Flexibility of some dynamical and geometrical data

We introduce the flexibility program proposed by A. Katok and discuss first results. First, we describe a flexibility result for Lyapunov exponents for smooth expanding maps on a circle of fixed degree. Then, we show the flexibility of the entropy with respect to the Liouville measure and topological entropy for geodesic flow on negatively curved surfaces with fixed genus and total area (joint with A. Katok). Also, we point out some restrictions which come from additionally fixing a conformal class of metrics (joint with T. Barthelm’e).