Ergodic theory is the branch of dynamical systems which aims to describe the asymptotic behavior of almost all orbits, with respect to invariant measures.  However there are situations in which such a description is rather trivial, as the case when the invariant measures are finite convex combinations of Dirac measures. The thermodynamic formalism in dynamical systems aims to construct invariant measures that are relevant with respect to some potential function and, in many cases, are physically relevant (meaning that these satisfy some Gibbs property). The theory is very well understood in the case of uniformly hyperbolic dynamical systems by the work of Sinai, Ruelle and Bowen.  In this seminar I will describe some recent  results on the thermodynamic formalism of random dynamical systems displaying some weak form of hyperbolicity.

This is a joint work with Shintaro Suzuki (Keio University) and Manuel Stadlbauer (Federal University of Rio de Janeiro).