Improved estimates of statistical properties in some non-uniformly hyperbolic dynamical systems
Péter Bálint, Ábel Komálovics
TL;DR
The paper develops a general scheme to tighten bounds on decay of correlations and the almost sure invariance principle for a broad class of non-uniformly hyperbolic systems with polynomial, summable rates. By extending Chernov-Markarian-Zhang and Young-tower frameworks and exploiting regularly varying tails, it aligns tower-return statistics with geometric return tails and eliminates logarithmic losses. For canonical examples like Wojtkowski’s two falling balls and Bunimovich flowers, it yields decay rates of the form $n^{-eta}$ (with $eta>1$) and improved ASIP rates, applicable to general and specialized observables. The results provide both upper bounds and asymptotics, with practical implications for rigorously quantifying mixing and limit laws in complex billiard-type systems.
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.