Abstract: The topological dynamics of foliations is well understood. There is an analogue of periodic orbits, limit sets, recurrence, and even topological entropy, following the work of Ghys, Langevin and Walczak. The topological entropy measures the transverse separation of the leaves of a foliation. This is not the case of the ergodic theory of foliations. In particular, there is no satisfactory version of metric entropy for foliations, and it is difficult to imagine a way to detect the separation of leaves using measure theory (whether harmonic, or invariant by certain dynamics tangent to the leaves). For example, is it possible to obtain a variational principle for foliations? In this talk, I will discuss an approach made with Sebastién ALVAREZ (Universidad de la República) to attack this problem.
