Escape Rates and Singular Limiting Distributions for Intermittent Maps with Holes

TL;DR

This paper investigates escape dynamics in intermittent maps with holes, revealing polynomial decay of surviving volume, dependence on initial distribution, and singular limiting measures, contrasting with exponential escape systems.

Contribution

It provides new insights into escape rates and limiting distributions for intermittent maps with holes, especially under subexponential escape conditions.

Findings

Surviving volume decays polynomially over time.

Normalized measures converge to a point mass at the origin.

Limiting measures are singular and supported on the survivor set.

Abstract

We study the escape dynamics in the presence of a hole of a standard family of intermittent maps of the unit interval with neutral fixed point at the origin (and finite absolutely continuous invariant measure). Provided that the hole (is a cylinder that) does not contain any neighborhood of the origin, the surviving volume is shown to decay at polynomial speed with time. The associated polynomial escape rate depends on the density of the initial distribution, more precisely, on its behavior in the vicinity of the origin. Moreover, the associated normalized push forward measures are proved to converge to the point mass supported at the origin, in sharp contrast to systems with exponential escape rate. Finally, a similar result is obtained for more general systems with subexponential escape rates; namely that the Ces\`aro limit of normalized push forward measures is typically singular, invariant and supported on the asymptotic survivor set.