Local Escape Rates for $\phi$-mixing Dynamical Systems

TL;DR

This paper investigates local escape rates in $\, extphi$-mixing dynamical systems, showing exponential rates at non-periodic points and extremal index at periodic points, with broad applications to various dynamical models.

Contribution

It establishes a general result linking local escape rates to $\, extphi$-mixing properties and extremal indices, extending to systems modeled by Young towers.

Findings

Escape rates are exponential with rate 1 at non-periodic points.

At periodic points, escape rates equal the extremal index.

Results apply to a wide class of dynamical systems including subshifts and interval maps.

Abstract

We show that dynamical systems with $\phi$-mixing measures have local escape rates which are exponential with rate $1$ at non-periodic points and equal to the extremal index at periodic points. We apply this result to equilibrium states on subshifts of finite type, expanding interval maps, Gibbs states on conformal repellers and more generally to Young towers and by extension to all systems that can be modeled by a Young tower.