Effective intrinsic ergodicity and the transport-entropy inequality: a relation and applications

Abstract

Effective intrinsic ergodicity (EIE) quantifies how the pressure defect of an invariant measure controls its distance from an equilibrium state. In this paper, we make explicit the connection between the Einsiedler-Kadyrov-Polo (EKP) inequality, which is central to EIE, and the transport-entropy inequality (TEI). We also extend the finite-state transport-entropy inequality of \cite{CMU19} to topologically mixing countable Markov shifts with the Big Images and Preimages (BIP) property, yielding an EIE estimate in Ornstein's $\bar d$-metric that strengthens \cite{Ruh21}. We also establish a local $\frac{1}{2}$-Hölder bound for the map sending a Hölder potential of finite pressure to its Bowen-Gibbs measure, with respect to the uniform norm on potentials and the $\bar d$-metric on measures. This result extends some aspects of Bhullar's continuity theorem \cite{B25}: it applies to proper subshifts, provides an explicit modulus of continuity with respect to the uniform norm, and does not require strong positive recurrence.