Estimates to correlation decay in discrete dynamical systems by spectral methods

We discuss the use of spectral methods to estimate the decay of correlations of observables in discrete dynamical systems. Some abstract results are stated and applied to the analysis of correlation decay for smooth and analytic observables considered on a particular class of skew-endomorphisms of the 2-torus. General superexponential estimates to correlation decay are also established for the algebraic automorphisms and endomorphisms of the n-torus, whenever hyperbolic or purely expanding conditions occur.