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In this work we study strong spectral properties of Ruelle transfer operatorsrelated to a large family of Gibbs measures for contact Anosov flows. Theultimate aim is to establish exponential decay of correlations for H\"olderobservables with respect to a very general class of Gibbs measures. Theapproach invented in 1997 by Dolgopyat \cite{D1} and further developed in\cite{St2} is substantially refined here, allowing to deal with much moregeneral situations than before, although we still restrict ourselves to theuniformly hyperbolic case. A rather general procedure is established whichproduces the desired estimates whenever the Gibbs measure admits a Pesin setwith exponentially small tails, that is a Pesin set whose preimages along theflow have measures decaying exponentially fast. We call such Gibbs measuresregular. Recent results in \cite{GSt} prove existence of such Pesin sets forhyperbolic diffeomorphisms and flows for a large variety of Gibbs measuresdetermined by H\"older continuous potentials. The strong spectral estimates forRuelle operators and well-established techniques lead to exponential decay ofcorrelations for H\"older continuous observables, as well as to some otherconsequences such as: (a) existence of a non-zero analytic continuation of theRuelle zeta function with a pole at the entropy in a vertical strip containingthe entropy in its interior; (b) a Prime Orbit Theorem with an exponentiallysmall error.

Spectral properties of Ruelle transfer operators for regular Gibbs measures and decay of correlations for contact Anosov flows