Regular and stochastic behaviour in dynamical systems

The focus of the proposal is the study of the long time behaviour in dynamical systems. This is a momentous problem to which the international community has dedicated a tremendous amount of work with an intensity that shows no sign of dwindling. Far-reaching results have been obtained, but limited to special systems: A) hyperbolic systems (for which the long time behaviour is stochastic in nature and hence naturally described in statistical terms); B) (piecewise algebraic or rigid) parabolic systems (which enjoy weaker ergodic properties characterised by the rate of convergence of the Birkhoff averages); C) perturbations of completely integrable systems (elliptic systems) whose ergodic properties are currently beyond our reach hence the emphasis is on the study of invariant sets. Our goal is to substantially forward the state of the art in all such cases. In case (A) we are concentrating on partially hyperbolic systems and hyperbolic systems with infinite invariant measure (e.g. Farey map) or non compact phase space (e.g. Lorentz gas). In (B) we aim at refining and extending known results beyond the rigid case. As for (C) we want to go beyond the symplectic context in order to make the techniques developed in the Hamiltonian formalism bear on a more general class of systems