Nonstandard functional central limit theorem for nonuniformly hyperbolic dynamical systems, including Bunimovich stadia

TL;DR

This paper proves a nonstandard functional central limit theorem for certain nonuniformly hyperbolic systems, including billiards and geodesic flows, with applications to Bunimovich stadia.

Contribution

It demonstrates that a CLT with nonstandard normalization implies a corresponding weak invariance principle for these systems, unifying and extending existing results.

Findings

WIP with nonstandard normalization holds for billiards and flows

Unified approach streamlines proofs in the literature

Bunimovich stadia satisfy WIP as a consequence of CLT

Abstract

We consider a class of nonuniformly hyperbolic dynamical systems with a first return time satisfying a central limit theorem (CLT) with nonstandard normalisation $(n\log n)^{1/2}$. For such systems (both maps and flows) we show that it automatically follows that the functional central limit theorem or weak invariance principle (WIP) with normalisation $(n\log n)^{1/2}$ holds for H\"older observables. Our approach streamlines certain arguments in the literature. Applications include various examples from billiards, geodesic flows and intermittent dynamical systems. In this way, we unify existing results as well as obtaining new results. In particular, we deduce the WIP with nonstandard normalisation for Bunimovich stadia as an immediate consequence of the corresponding CLT proved by B\'alint & Gou\"ezel.