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.

References in corpus (1)

• Strong invariance principles with rate for "reverse" martingales and applications

Cited by in corpus (4)

• Rates in almost sure invariance principle for dynamical systems with some hyperbolicity
• Sharp Statistical Properties for a Family of Multidimensional NonMarkovian Nonconformal Intermittent Maps
• Non-stationary Almost Sure Invariance Principle for Hyperbolic Systems with Singularities
• On Coupling Lemma and Stochastic Properties with Unbounded Observables for 1-d Expanding Maps