Equidistribution for Nonuniformly Expanding Dynamical Systems, and Application to the Almost Sure Invariance Principle

Abstract

Let ${T : M \to M}$ be a nonuniformly expanding dynamical system, such as logistic or intermittent map. Let ${v : M \to \mathbb{R}^d}$ be an observable and ${v_n = \sum_{k=0}^{n-1} v \circ T^k}$ denote the Birkhoff sums. Given a probability measure ${\mu}$ on M, we consider v _n as a discrete time random process on the probability space ${(M, \mu)}$. In smooth ergodic theory there are various natural choices of ${\mu}$, 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.