We consider a discrete dynamical system f : M → M, where M is a Riemannian manifold and f is a diffeomorphism. We assume that the dynamical system has a Gibbs- Markov-Young structure, which consists of a reference set Λ with a hyperbolic product structure that satisfies certain properties. The properties assumed here are the existence of a Markov partition Λ1, Λ2, . . . of Λ, polynomial contraction on stable leaves, polynomial backwards contraction on unstable leaves, a bounded distortion property and a certain regularity of the stable foliation.

Our main goals are to prove results establishing a control on the decay of correlations and large deviations, as well as presenting an example of a dynamical system satisfying the Gibbs-Markov-Young structure described above.