Volume lemmas for partially hyperbolic endomorphisms

In this talk, we discuss contributions to the thermodynamic formalism of partially hyperbolic attractors for non-singular endomorphisms. We consider a class of local diffeomorphisms admitting an invariant stable bundle and a positively invariant cone field with non-uniform cone expansion at a positive Lebesgue measure set of points. We present how to construct SRB measures for this class of maps and prove estimates for the measure of dynamical balls with respect to the volume measure and the SRB measure. As an application of these estimates, we obtain large deviations bounds for the convergence of Birkhoff ergodic averages. This is a joint work with Paulo Varandas (UFBA-BR e UP-PT) and Giovane Ferreira (UFMA-BR).