Physical measures for the foliated geodesic flow
Sébastien Alvarez (IMPA)
In this talk we will study the ergodic properties of the geodesic ow tangent to the leaves of a foliation, say, of codimension 1 when the leaves are negatively curved. Such a ow exhibits a weak form of hyperbolicity called foliated hyperbolicity, due to the curvature assumption. It resembles the classical notion of partial hyperbolicity: the transverse direction of the foliation playing the role of the central direction. Howerver there is a main diference since the dynamics in this direction is not a priori dominated by the hyperbolicity in the foliation. We will show a dichotomy.
- Either there is a transverse holonomy-invariant measure;
- Or the foliation has nitely many minimal sets. In that case each of these minimal sets supports a unique SRB measure for the foliated geodesic ow, whose transverse Lyapunov exponent is negative. And their basins cover a full volume set.
We will also focus on special examples: foliations transverse to a projective circle bundle over a hyperboilc surface and show that a simple topological condition (about Euler number of the bundle) ensures that the foliated geodesic ow is partially hyperbolic, leading to new geometric examples of partially hyperbolic ows. This is a joint work with J.Yang