Escape rate scaling in infinite measure preserving systems

TL;DR

This paper studies how the rate at which points escape from certain intermittent dynamical systems scales with the size of a hole, revealing logarithmic and polynomial behaviors in infinite measure cases.

Contribution

It introduces a detailed analysis of escape rate scaling in infinite measure-preserving systems with intermittency and proposes a conjecture linking wandering rate to escape dynamics.

Findings

Logarithmic corrections to escape rate scaling observed

Polynomial scaling of escape rate with hole size identified

Conjecture relating wandering rate to escape rate scaling proposed

Abstract

We investigate the scaling of the escape rate from piecewise-linear dynamical systems displaying intermittency due to the presence of an indifferent fixed-point. Strong intermittent behaviour in the dynamics can result in the system preserving an infinite measure. We define a neighbourhood of the indifferent fixed point to be a hole through which points escape and investigate the scaling of the rate of this escape as the length of the hole decreases, both in the finite measure preserving case and infinite measure preserving case. In the infinite measure preserving systems we observe logarithmic corrections to and polynomial scaling of the escape rate with hole length. Finally we conjecture a relationship between the wandering rate and the observed scaling of the escape rate.