This project deals with some aspects of the qualitative and statistical theory of dynamical systems. Probability is a natural tool to understand dynamical systems. From one side, stochastic processes are universal models of many physical systems. From the other side, most interesting deterministic systems are best understood from the point of view of ergodic theory.
Our main concerns are: limit theorems for sequences of associated variables or for branching processes, stochastic stability of Markov chains, decay of correlations. Sequences of associated variables, first introduced as examples of weakly dependent stochastic processes, also appear naturally in statistical mechanics as models of ferromagnetic or anti-ferromagnetic spin systems. A basic physical issue about deterministic systems is their stability under some sort of perturbations. Markov chains, beside their importance as models, are natural tools, as well as sufficiently tractable examples, to understand generic and stable properties.