Cover time for countable Markov shifts

TL;DR

This paper investigates the cover time in countable Markov shifts with Gibbs measures, showing it depends on the minimal measure of metric balls and is sensitive to metric changes.

Contribution

It establishes a relationship between cover time and minimal measure of balls in countable Markov shifts, highlighting metric sensitivity.

Findings

Cover time relates to the minimum measure of metric balls.

Cover time behavior is sensitive to metric modifications.

Results apply to countable full shifts with Gibbs measures.

Abstract

Cover time, in the context of dynamical systems, quantifies the rate at which orbits cover the system. We prove that for countable full shifts with a Gibbs measure, equipped with a natural metric, the rate of covering of orbits of points behaves according to the minimum measure of balls. Moreover, this rate exhibits sensitivity to changes in the metric.