Linear drift and entropy for regular covers

We consider a regular Riemannian cover \M of a compact Riemannian manifold. The linear drift $\ell$ and the Kaimanovich entropy $h$ are geometric invariants defined by asymptotic properties of the Brownian motion on \M. We show that $\ell^2 \leq h$

Volume entropy rigidity of non-positively curved symmetric spaces

We characterize symmetric spaces of non-positive curvature by the equality case of general inequalities between geometric quantitie

Ergodic properties of equilibrium measures for smooth three dimensional flows

Let $\{T^t\}$ be a smooth flow with positive speed and positive topological entropy on a compact smooth three dimensional manifold, and let $\mu$ be an ergodic measure of maximal entropy. We show that either $\{T^t\}$ is Bernoulli, or $\{T^t\}$ is isomorphic to the product of a Bernoulli flow and a rotational flow. Applications are given to Reeb flows.Comment: 32 pages, 1 figure, a section on equilibrium measures for multiples of the geometric potential has been added, to appear in Commentarii Mathematici Helvetic

Differentiating the stochastic entropy for compact negatively curved spaces under conformal changes

We consider the universal cover of a closed Riemannian manifold of negative sectional curvature. We show that the linear drift and the stochastic entropy are differentiable under any C^3 one-parameter family of C^3 conformal changes of the original metric.Comment: Revised versio

Entropy rigidity of symmetric spaces without focal points

We characterize symmetric spaces without focal points by the equality case of general equalities between geometric quantities.Comment: The proof of Theorem 1.2 in the previous versions rested on a statement in [CFL], the proof of which is incomplete. Theorem 1.2 has been removed from this last versio