Hitting statistics for $\phi$-mixing dynamical systems

TL;DR

This paper establishes a general limit law connecting hitting and escape rates in $\,\,\phi$-mixing dynamical systems, with explicit formulas for extremal indices, applicable to systems like hyperbolic and expanding maps.

Contribution

It extends recurrence law analysis to $\,\,\phi$-mixing systems, providing explicit extremal index formulas and broad applicability to systems with Young towers.

Findings

The limit equals one for systems without short returns.

The limit is less than one when short returns occur.

Explicit extremal index formulas are derived.

Abstract

Hitting rate and escape rate are two examples of recurrence laws for a dynamical system, and a general limit connects them. We show that for both Gibbs-Markov systems or any systems with the $\phi$-mixing measure, for a sequence of nested sets whose intersection is a measure zero set, this general limit equals one in the absence of short returns and less than one otherwise, which is given by an explicit formula called extremal index. One of the applications of this result is to dynamical systems on Riemannian manifolds such as hyperbolic maps and expanding maps, and it can be applied to any system with a suitable Young tower.