Aula Riunioni, Dipartimento di Matematica
The modern theory of Dynamical Systems aims to describe the asymptotic behavior and complexity of maps and flows, with the intervention of topological, geometric and probabilistic methods. Inspired by ideas from statistical mechanics, a first objective in the thermodynamic formalism in dynamical systems is to establish a connection – referred to as variational principle – between the topological complexity of the dynamics and the complexity exhibited by invariant measures, and to study the existence and statistical properties of measures of maximal complexity (called equilibrium states). Such a program was carried out much successfully in the seventies by Sinai, Ruelle and Bowen in the context of uniformly hyperbolic dynamical systems. The general picture is however far from being completely understood.
In this talk I will recall some classical results about the thermodynamic formalism of uniformly hyperbolic maps and discuss some of the current challenges of extending the theory beyond this context.