Martingale Methods for Maximal Large Deviations and Young Towers

TL;DR

This paper introduces martingale approximation techniques to derive quantitative maximal large deviations estimates for invertible dynamical systems, with applications to Young towers and partially hyperbolic diffeomorphisms.

Contribution

It develops a new martingale framework that connects decay of correlations to large deviations, extending results to systems modeled by Young towers with subexponential behavior.

Findings

Derived maximal large deviations estimates for invertible dynamical systems.

Established Young structures with matching recurrence tails for certain diffeomorphisms.

Applied results to slowly mixing billiard systems with subexponential properties.

Abstract

We develop a martingale approximation framework yielding quantitative maximal large deviations estimates for invertible dynamical systems. From suitable decay of correlations, we deduce these estimates and, as an application, we obtain Young structures with matching recurrence tails for partially hyperbolic diffeomorphisms with mostly expanding central direction. In a second application, we prove maximal large deviation estimates for systems modelled by Young towers with subexponential contraction and expansion. Many examples of slowly mixing billiards are covered by this result.