Optimal bounds for decay of correlations and {\alpha}-mixing for   nonuniformly hyperbolic dynamical systems
  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. 
Sandro Vaienti, Hong-Kun Zhang
math.DS