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We investigate the decay rates of correlations for nonuniformly hyperbolicsystems with or without singularities, on piecewise H\"older observables. Byconstructing a new scheme of coupling methods using the probability renewaltheory, we obtain the optimal bounds for decay rates of correlations for alarge class of such observables. We also establish the alpha-mixing propertyfor time series generated by these systems, which leads to a vast ranges oflimiting theorems. Our results apply to rather general hyperbolic systems withsingularities, including Bunimovich flower billiards, semidispersing billiardson a rectangle and billiards with cusps, and other nonuniformly hyperbolicmaps.

Optimal bounds for decay of correlations and {\alpha}-mixing for nonuniformly hyperbolic dynamical systems