  We investigate the decay rates of correlations for nonuniformly hyperbolic
systems with or without singularities, on piecewise H\"older observables. By
constructing a new scheme of coupling methods using the probability renewal
theory, we obtain the optimal bounds for decay rates of correlations for a
large class of such observables. We also establish the alpha-mixing property
for time series generated by these systems, which leads to a vast ranges of
limiting theorems. Our results apply to rather general hyperbolic systems with
singularities, including Bunimovich flower billiards, semidispersing billiards
on a rectangle and billiards with cusps, and other nonuniformly hyperbolic
maps.
