Rates for maps and flows in a deterministic multidimensional weak invariance principle

TL;DR

This paper establishes the first convergence rates to multidimensional Brownian motion for various dynamical systems, including continuous time cases and nonuniformly hyperbolic systems.

Contribution

It provides novel convergence rates for multidimensional invariance principles in both discrete and continuous time dynamical systems.

Findings

First rates of convergence for N-dimensional Brownian motion in discrete and continuous time.

Applicable to nonuniformly hyperbolic and expanding systems like Axiom A flows.

Includes continuous time in any dimension, a first in the field.

Abstract

We present the first rates of convergence to an $N$-dimensional Brownian motion when $N\ge2$ for discrete and continuous time dynamical systems. Additionally, we provide the first rates for continuous time in any dimension. Our results hold for nonuniformly hyperbolic and expanding systems, such as Axiom A flows, suspensions over a Young tower with exponential tails, and some classes of intermittent solenoids.