Smale's Axiom A is a hyperbolicity assumption which implies complicated, "chaotic" (but now well-understood) behaviour for a discrete or continuous dynamical system. In this thesis, we study ergodic properties of some dynamical systems which do not satisfy Axiom A but which have analogous properties and we obtain results similar to the ones in the Axiom A case.

The thesis is divided in two independent parts.

In the first part, we study the analytic properties of weighted zeta functions and Fourier transforms of correlation functions for one-dimensional piecewise monotone dynamical systems and their suspensions (working on spaces of functions with bounded variation). Using previous results of Hofbauer, Keller and the author, we obtain results analogous to those due to Pollicott, Ruelle and Haydn in the Axiom A setting. We show how these results can be used to study the geometric Lorenz model and give an example of a tent map with complex "resonances".

In the second part, we study the finitely presented dynamical systems recently introduced by Fried and show that the notions of equilibrium states and Gibbs states (for Holder continuous functions) are equivalent. Our results extend those of Ruelle, Haydn and others on Axiom A systems.