Dynamics, smooth rigidity and ergodic properties of hyperbolic maps and flows

Abstract

In this project, we address three main lines of investigation: 1) perturbation of Anosov diffeomorphisms to remove resonances of Lyapunov exponents;2) to study properties of Anosov endomorphisms, such as smooth rigidity and existence of invariant geometric structures;3) ergodic properties and existence of Blenders for partially hyperbolic flows. In item (2), we conjecture that special Anosov endomorphisms for which the lifting to the universal covering admit smooth Anosov sub-bundles, are smoothly conjugated to a linear Anosov endomorphism. For the non-special case, we do not conjecture to obtain a specific type of result, although we also believe some kind of rigidity should occur. In item (1), the main goal is to develop tools to remove Lyapunov exponents resonances without changing the regularity of the stable and unstable Anosov sub-bundles. In this light, we aim to approach the problem of determining which manifolds admit Anosov diffeomorphisms with smooth splitting. In item (3), which is currently being developed jointly with Y. Arthur de Jesus and M. E. Noriega, we are working in the creation of Blender type structures for perturbed partially hyperbolic flows. These structures will be used to obtain ergodicity criteriums for generic families of flows. All the above-mentioned lines are connected to important problems in dynamical systems and, more specifically, in smooth ergodic theory. (AU)