Hitting and escaping statistics: mixing, targets and holes

TL;DR

This paper explores the relationship between hitting times to shrinking targets and escape times in dynamical systems, revealing a phase transition based on the system's mixing speed.

Contribution

It establishes a connection between hitting time laws and escape laws, showing how this relationship depends on the system's mixing rate.

Findings

Exponential mixing systems allow transition between hitting and escape laws.

Subexponential mixing systems exhibit a phase transition between the two laws.

The results unify understanding of recurrence and escape phenomena in dynamical systems.

Abstract

There is a natural connection between two types of recurrence law: hitting times to shrinking targets, and hitting times to a fixed target (usually seen as escape through a hole). We show that for systems which mix exponentially fast, one can move through a natural parameter space from one to the other. On the other hand, if the mixing is subexponential, there is a phase transition between the hitting times law and the escape law.