Stochastic properties of dynamical systems

Some outstanding open problems in the field of Dynamical systems are: the derivation of precise limit laws for smooth hyperbolic systems, the study of partially hyperbolic systems, discontinuous systems, extended systems, systems with many, possibly infinitely many, degrees of freedom, and systems with mixed phase space. Our objective is to address the above-mentioned momentous problems by investigating general systems and specific models that we believe will shed light on general issues. While our final goals are highly ambitious, and hence very risky, all the intermediate achievements of our project are of great interest in themselves. Therefore, any partial success in this research program would constitute a significant advance of the state of the art. In addition, as we detail shortly, many tasks are interconnected, so any advance in one will facilitate advances in the others. More precisely, we intend to better study the spectral theory of transfer operators on functional spaces whose topology encodes the geometric properties of the system. This task (TO) is subdivided into six subtasks TO1) the study of the dynamical zeta function; TO2) the study of refined limit laws, with small errors, for observable of hyperbolic dynamical systems; TO3) the study of partially hyperbolic systems; TO4) the study of renormalizable parabolic systems; TO6) the study of mean-field systems and the derivation of Vlasov-like dynamics in billiard systems. The next goal is the study of extended (non-compact) systems or systems with respect to a sigma-finite measure. This task (NC) is subdivided into five subtasks NC1) the study of Birkhoff sums; NC2) the study of infinite mixing; NC3) the study of horocycle flow and multi-dimensional continued fractions; NC4) the study of thermodynamic formalism for infinite-measure-preserving intermittent maps; NC5) the study of the Lorentz gas, especially the aperiodic case. Finally, we would like to investigate symplectic systems with both regular and chaotic behavior. Indeed, the case in which a system displays both regular (KAM island) and chaotic (positive metric entropy) behaviour is believed to be typical. This task (MP) is subdivided into three subtasks MP1) the study of homoclinic hyperbolic orbits (and hence positive topological entropy) in billiards; MP2) the study of systems with some randomness, e.g., billiards with random reflection law, to address the problem of the size of the Lyapunov exponents and mixing; MP3) the study of the precise speed of the convergence to equilibrium of a standard map with very small randomness.