Effective intrinsic ergodicity for countable state Markov shifts

TL;DR

This paper establishes bounds on the distance between invariant measures and maximal entropy measures for strongly positively recurrent countable state Markov shifts, extending previous finite type results and providing sharp bounds for measures with near-maximal entropy.

Contribution

It extends bounds on measure distances from finite type to countable state Markov shifts and introduces sharp bounds for measures close to maximal entropy.

Findings

Bound on measure distance in terms of entropy difference

Extension of Kadyrov's finite type results to countable shifts

Sharp bounds for measures with near-maximal entropy

Abstract

For strongly positively recurrent countable state Markov shifts, we bound the distance between an invariant measure and the measure of maximal entropy in terms of the difference of their entropies. This extends an earlier result for subshifts of finite type, due to Kadyrov. We provide a similar bound for equilibrium measures of strongly positively recurrent potentials, in terms of the pressure difference. For measures with nearly maximal entropy, we have new, and sharp, bounds. The strong positive recurrence condition is necessary.