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 $\ell^2 \leq h$

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

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

Singularity of projections of 2-dimensional measures invariant under the geodesic flow

We show that on any compact Riemann surface with variable negative curvature there exists a measure which is invariant and ergodic under the geodesic flow and whose projection to the base manifold is 2-dimensional and singular with respect to the 2-dimensional Lebesgue measure.Comment: 12 page