Large deviations and rates of convergence in the Birkhoff ergodic theorem: From Hölder continuity to continuity

Abstract

It is established that, for ergodic dynamical systems, upper estimates for the decay of large deviations of ergodic averages for (non-Holder) continuous almost everywhere averaged functions have the same asymptotics as in the Holder continuous case. The results are applied to obtaining the corresponding estimates for large deviations and rates of convergence in the Birkhoff ergodic theorem with non-Holder averaged functions in certain popular chaotic billiards, such as the Bunimovich stadiums and the planar periodic Lorentz gas.