A Vector-Valued Almost Sure Invariance Principle for Hyperbolic Dynamical Systems

TL;DR

This paper establishes an almost sure invariance principle for vector-valued observables in nonuniformly hyperbolic dynamical systems, showing they can be approximated by Brownian motion, including applications to the planar Lorentz gas.

Contribution

It extends the invariance principle to a broad class of hyperbolic systems, including Axiom A diffeomorphisms and flows, and systems modeled by Young towers.

Findings

Vector-valued observables approximate Brownian motion.

Applicable to Axiom A systems and Young towers.

Planar Lorentz gas position variable approximates 2D Brownian motion.

Abstract

We prove an almost sure invariance principle (approximation by d-dimensional Brownian motion) for vector-valued Holder observables of large classes of nonuniformly hyperbolic dynamical systems. These systems include Axiom~A diffeomorphisms and flows as well as systems modelled by Young towers with moderate tail decay rates. In particular, the position variable of the planar periodic Lorentz gas with finite horizon approximates a 2-dimensional Brownian motion.