  We consider diffeomorphisms of compact Riemmanian manifolds which have a
Gibbs-Markov-Young structures, consisting of a reference set $\Lambda$ with a
hyperbolic product structure and a countable Markov partition. We assume
polynomial contraction on stable leaves, polynomial backward contraction on
unstable leaves, a bounded distortion property and a certain regularity of the
stable foliation. We establish a control on the decay of correlations and large
deviations of the SRB measure of the dynamical system, based on a polynomial
control on the Lebesgue measure of the tail of return times. Finally, we
present an example of a dynamical system defined on the torus and prove that it
verifies the properties of the Gibbs-Markov-Young structure that we considered.
