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

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## 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.

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