Correlation Decay and Recurrence Asymptotics for Some Robust Nonuniformly Hyperbolic Maps

Abstract. We study decay of correlations, the asymptotic distribution of hitting times and fluctuations of the return times for a robust class of multidimensional non-uniformly hyperbolic transformations. Oliveira and Viana [15] proved that there is a unique equilibrium state μ for a large class of nonuniformly expanding transformations and Hölder continuous potentials with small variation. For an open class of potentials with small variation, we prove quasi-compactness of the Ruelle-Perron-Frobenius operator in a space Vθ of functions with essential bounded variation that strictly contain Hölder continuous observables. We deduce that the equilibrium states have exponential decay of correlations. Furthermore, we prove exponential asymptotic distribution of hitting times and log-normal fluctuations of the return times around the average hμ(f).