Improved estimates of statistical properties in some non-uniformly hyperbolic dynamical systems

TL;DR

This paper introduces a new method to improve bounds on statistical properties like decay of correlations in certain non-uniformly hyperbolic dynamical systems, achieving essentially optimal results by removing logarithmic factors.

Contribution

The authors develop a general scheme that enhances previous bounds on statistical properties for non-uniformly hyperbolic systems with polynomial mixing rates, removing logarithmic factors.

Findings

Achieved essentially optimal bounds on decay of correlations.

Applied method to systems like falling balls and dispersing billiards.

Removed logarithmic factors from previous estimates.

Abstract

Building upon previous works by Young, Chernov-Zhang and Bruin-Melbourne-Terhesiu, we present a general scheme to improve bounds on the statistical properties (in particular, decay of correlations, and rates in the almost sure invariant principle) for a class of non-uniformly hyperbolic dynamical systems. Specifically, for systems with polynomial, yet summable mixing rates, our method removes logarithmic factors of earlier arguments, resulting in essentially optimal bounds. Applications include Wojtkowski's system of two falling balls, dispersing billiards with flat points and Bunimovich's flower-shaped billiard tables.