Sébastien Alvarez et Christian Bonatti

Foliated hyperbolicity and ergodic theory for foliations.

Abstract: In general foliations do not admit any invariant measures. This does not means that there is no ergodic theory for foliations.

  • One possibility is to consider harmonic measures, which are in some sense the equivalent of stationnary measures for group actions
  • Another strategy consists in considering the flow which is the geodesic flow in restriction to each the leaf. In the case of negatively curved leaves, this flow is hyperbolic in the leaves. This motivates the notion of foliated hyperbolicity and we will consider some cases where this notion leads to a notion of attractors for foliations. In contrast with the usual dynamics, a same leaf will visit several attractors: for a given point there is a proportion of directions in which the leaf is attracted by a given attractor.
  • In the specific case of foliations obtained by suspension, we will see that the ergodic theory of the geodesic flow, in particular the Gibbs states associated to several potentials, lead to measures which describe (according to the potential) the geodesics, the brownian motion (harmonic measures) or the large discs in the leaves (related with Margulis measure).
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