Escape rates for the Farey map with approximated holes

TL;DR

This paper investigates how the escape rate in the Farey map, an infinite measure system with a hole near an indifferent fixed point, varies with different approximations of the hole, revealing convergence to known behaviors.

Contribution

The study introduces a novel approximation method for analyzing escape rates in the Farey map with a hole near the indifferent fixed point, overcoming ergodic property challenges.

Findings

Escape rate scaling depends on the approximation method used.

Numerical results show convergence to known behaviors from piecewise linear approximations.

Approximation approach enables analysis where standard methods fail.

Abstract

We study the escape rate for the Farey map, an infinite measure preserving system, with a hole including the indifferent fixed point. Due to the ergodic properties of the map, the standard theoretical approaches to this problem cannot be applied. To overcome this difficulties we propose here to consider approximations of the hole by means of real analytic functions. We introduce a particular family of approximations and study numerically the behavior of the escape rate for "shrinking" approximated holes. The results suggest that the scaling of the escape rate depends on the chosen approximation, but "converges" to the behavior found for piecewise linear approximations of the map in \cite{KM}.