I will discuss how to obtain Young towers with exponential tails for multi-dimensional piecewise uniformly expanding maps of R^N.

We will discuss 3-dimensional parabolic flows which are renormalised by circle extensions of Anosov diffeormorphisms (this includes nilflows on the Heisenberg nilmanifold). We use the spectral information of the transfer operators associated to these partially hyperbolic maps to describe the deviation of ergodic averages and solutions of the cohomological equation for the parabolic flow.

We prove the uniqueness of the measure of maximal entropy (MME) for the geodesic flows on surfaces without conjugate points. We also prove that this unique MME is fully supported, can be obtained from limiting distrobution along closed orbits and the flow is mixing with respect to it. As a consequence we have that the closed orbits are dense in the unit tangent bundle. We also formulate conditions under which these results generalise to higher dimension.
 

Everyone is welcome to attend