|In this talk we address ourselfs to some recent results (joint with M.
Viana) about existence, uniqueness and some ergodic properties of
equilibrium measures for local diffeomorphisms exibting non-uniform
expanding behaviour. Two main results will
a) Existence and uniqueness of maximal entropy measures;
b) Uniqueness of equilibrium measures for H\"older continuous
potentials not too far from constant.
In this last case, we develop a Ruelle-Perron-Fr\"obenius transfer
to the ergodic theory of a large class of non-uniformly expanding
transformations on compact manifolds. We prove that the transfer
operator has a positive eigenfunction, piecewise H\"older
continuous, and use this fact to show that there is exactly one
equilibrium state. Moreover, the equilibrium state is a non-lacunary
Gibbs measure, a non-uniform version of the classical notion of
Time permiting, we discuss some very recent extensions of these results