On the escape rate for intermittent maps with holes shrinking around the indifferent fixed point

TL;DR

This paper investigates how the escape rate behaves in non-uniformly expanding interval maps with a parabolic fixed point when a shrinking hole around that point is introduced, extending previous specific case studies.

Contribution

It introduces a general framework for analyzing escape rates in maps with indifferent fixed points, using transfer operators and inducing techniques, broadening prior specialized results.

Findings

Derived asymptotic formulas for escape rates as holes shrink.

Extended previous results to more general non-uniformly expanding maps.

Connected transfer operators of original and induced systems.

Abstract

We study non-uniformly expanding maps of the unit interval with a parabolic fixed point at the origin that admit an ergodic absolutely continuous invariant measure, which may be finite or infinite. By introducing a hole defined by an interval containing the parabolic fixed point, we analyze the escape rate of the resulting open system and its asymptotic behavior as the hole shrinks. Our approach relies on the transfer operator associated with the dynamical system and on the relationship between the transfer operators of the original system and its induced version. The results extend to this general framework previous investigations which considered special cases.