Spectrum and statistical properties of chaotic dynamics

We present new developments on the statistical properties of chaotic dynamical systems. We concentrate on the existence of an ergodic physical (SRB) invariant measure and its mixing properties, in particular decay of its correlation functions for smooth observables. In many cases, there is a connection (via the spectrum of a Ruelle-Perron-Frobenius transfer operator) with the analytic properties of a weighted dynamical zeta function, weighted dynamical Lefschetz function, or dynamical Ruelle-Fredholm determinant, built using the periodic orbit structure of the map.