A note on statistical properties for nonuniformly hyperbolic systems with slow contraction and expansion

TL;DR

This paper develops a systematic method using martingale-coboundary decomposition to establish statistical limit laws for nonuniformly hyperbolic systems with slow contraction and expansion, including CLT and invariance principles.

Contribution

It introduces a unified approach for deriving limit laws in nonuniformly hyperbolic systems with slow dynamics, extending previous results to broader classes.

Findings

Establishes CLT for systems with square-integrable return times.

Proves weak invariance principle and its iterated version.

Provides a framework for statistical analysis of complex dynamical systems.

Abstract

We provide a systematic approach for deducing statistical limit laws via martingale-coboundary decomposition, for nonuniformly hyperbolic systems with slowly contracting and expanding directions. In particular, if the associated return time function is square-integrable, then we obtain the central limit theorem, the weak invariance principle, and an iterated version of the weak invariance principle.