Rate of Convergence in the Weak Invariance Principle for Deterministic Systems

TL;DR

This paper establishes the first convergence rate results in the Prokhorov metric for the weak invariance principle in various deterministic dynamical systems, including hyperbolic and nonuniformly hyperbolic types.

Contribution

It provides novel convergence rate results in the Prokhorov metric for the functional central limit theorem in deterministic systems, extending to complex nonuniformly hyperbolic cases.

Findings

Convergence rates are established for hyperbolic systems.

Results apply to nonuniformly hyperbolic systems like billiards and Henon maps.

Applications include deterministic homogenization in multiscale systems.

Abstract

We obtain the first results on convergence rates in the Prokhorov metric for the weak invariance principle (functional central limit theorem) for deterministic dynamical systems. Our results hold for uniformly expanding/hyperbolic (Axiom A) systems, as well as nonuniformly expanding/hyperbolic systems such as dispersing billiards, Henon-like attractors, Viana maps and intermittent maps. As an application, we obtain convergence rates for deterministic homogenization in multiscale systems.