Poisson Limit Theorems for Systems with Product Structure

TL;DR

This paper establishes a Poisson limit law for return times in product dynamical systems, requiring only one hyperbolic factor and allowing the other to be elliptic or parabolic, with applications to Anosov maps.

Contribution

It introduces new Poisson limit theorems for product systems with minimal hyperbolic assumptions, extending to skew products and high-rank maps.

Findings

Poisson limit holds for systems with one hyperbolic factor

Applicable to Anosov maps and their products

Methods extend to certain skew products

Abstract

We obtain a Poisson Limit for return times to small sets for product systems. Only one factor is required to be hyperbolic while the second factor is only required to satisfy polynomial deviation bounds for ergodic sums. In particular, the second fact can be either elliptic or parabolic. As an application of our main result, several maps of the form Anosov map $\times$ another map are shown to satisfy a Poisson Limit Theorem at typical points, some even at all points. The methods can be extended to certain types of skew products, including $T,T^{-1}$-maps of high rank.