Equidistribution for nonuniformly expanding dynamical systems, and application to the almost sure invariance principle

Abstract

Let T: M→ M be a nonuniformly expanding dynamical system, such as logistic or intermittent map. Let v: M→ R d be an observable and vn=∑k=0n-1v∘Tk denote the Birkhoff sums. Given a probability measure μ on M, we consider v n as a discrete time random process on the probability space (M, μ). In smooth ergodic theory there are various natural choices of μ, such as the Lebesgue measure, or the absolutely continuous T-invariant measure. They give rise to different random processes. We investigate relation between such processes. We show that in a large class of measures, it is possible to couple (redefine on a new probability space) every two processes so that they are almost surely close to each other, with explicit estimates of “closeness”. The purpose of this work is to close a gap in the proof of the almost sure invariance principle for nonuniformly hyperbolic transformations by Melbourne and Nicol.