Quasi-Invariant measures, escape rates and the effect of the hole

TL;DR

This paper investigates how the size of a hole in a piecewise expanding interval map affects the existence of an absolutely continuous conditionally invariant measure with a bounded escape rate, using Ulam's method and perturbation theory.

Contribution

It provides a method to estimate the maximum hole size ensuring the existence of an accim with a specified escape rate in perturbed interval maps.

Findings

Derived an upper bound on hole size for accim existence

Applied Ulam's method and Keller-Liverani perturbation theory

Quantified escape rates in perturbed dynamical systems

Abstract

Let $T$ be a piecewise expanding interval map and $T_H$ be an abstract perturbation of $T$ into an interval map with a hole. Given a number $\ell$, $0<\ell<1$, we compute an upper-bound on the size of a hole needed for the existence of an absolutely continuous conditionally invariant measure (accim) with escape rate not greater than $-\ln(1-\ell)$. The two main ingredients of our approach are Ulam's method and an abstract perturbation result of Keller and Liverani.