Non-stationary Almost Sure Invariance Principle for Hyperbolic Systems with Singularities

TL;DR

This paper proves an almost sure invariance principle for a broad class of two-dimensional hyperbolic systems with singularities, including Anosov diffeomorphisms and Sinai billiards, even with unbounded observables and non-stationary processes.

Contribution

It establishes the ASIP for non-stationary, unbounded observables in hyperbolic systems with singularities, extending previous results to more general settings.

Findings

ASIP holds for hyperbolic systems with singularities.

Applicable to unbounded and non-stationary observables.

Includes systems like Sinai billiards and Anosov diffeomorphisms.

Abstract

We investigate a wide class of two-dimensional hyperbolic systems with singularities, and prove the almost sure invariance principle (ASIP) for the random process generated by sequences of dynamically H\"older observables. The observables could be unbounded, and the process may be non-stationary and need not have linearly growing variances. Our results apply to Anosov diffeomorphisms, Sinai dispersing billiards and their perturbations. The random processes under consideration are related to the fluctuation of Lyapunov exponents, the shrinking target problem, etc.