Criteria for the density of the graph of the entropy map restricted to ergodic states

TL;DR

This paper investigates conditions under which the entropy map's graph, restricted to ergodic states, is dense, providing criteria involving differentiability, large deviation principles, and convexity, with implications for equilibrium states.

Contribution

It introduces new criteria for the density of the entropy map graph in ergodic states, linking differentiability, large deviations, and convexity properties.

Findings

Criteria based on Gateaux differentiability of the pressure map.

Equivalence between density of the entropy graph and existence of a dense space of functions with unique equilibrium states.

Conditions involving large deviation principles and convexity properties.

Abstract

We consider a non-uniquely ergodic dynamical system given by a $\mathbb{Z}^{l}$-action (or $(\N\cup\{0\})^l$-action) $\tau$ on a non-empty compact metrisable space $\Omega$, for some $l\in\N$. Let (D) denote the following property: The graph of the restriction of the entropy map $h^\tau$ to the set of ergodic states is dense in the graph of $h^\tau$. We assume that $h^\tau$ is finite and upper semi-continuous. We give several criteria in order that (D) holds, each of which is stated in terms of a basic notion: Gateaux differentiability of the pressure map $P^\tau$ on some sets dense in the space $C(\Omega)$ of real-valued continuous functions on $\Omega$, level-2 large deviation principle, level-1 large deviation principle, convexity properties of some maps on $\R^n$ for all $n\in\N$. The one involving the Gateaux differentiability of $P^\tau$ is of particular relevance in the context of large deviations since it establishes a clear comparison with another well-known sufficient condition: We show that for each non-empty $\sigma$-compact subset $\Sigma$ of $C(\Omega)$, (D) is equivalent to the existence of an infinite dimensional vector space $V$ dense in $C(\Omega)$ such that $f+g$ has a unique equilibrium state for all $(f,g)\in \Sigma\times V\setminus\{0\}$; any Schauder basis $(f_n)$ of $C(\Omega)$ whose linear span contains $\Sigma$ admits an arbitrary small perturbation $(h_n)$ so that one can take $V=\textnormal{span}(\{f_n+h_n: n\in\N\})$. Taking $\Sigma=\{0\}$, the existence of an infinite dimensional vector space dense in $C(\Omega)$ constituted by functions admitting a unique equilibrium state is equivalent to (D) together with the uniqueness of measure of maximal entropy.