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

Abstract

We consider a non-uniquely ergodic dynamical system given by a$\mathbb{Z}^{l}$-action (or$(\mathbb{N}\cup \{0\})^{l}$-action)$\unicode[STIX]{x1D70F}$on a non-empty compact metrisable space$\unicode[STIX]{x1D6FA}$, for some$l\in \mathbb{N}$. Let (D) denote the following property: the graph of the restriction of the entropy map$h^{\unicode[STIX]{x1D70F}}$to the set of ergodic states is dense in the graph of$h^{\unicode[STIX]{x1D70F}}$. We assume that$h^{\unicode[STIX]{x1D70F}}$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$P^{\unicode[STIX]{x1D70F}}$on some sets dense in the space$C(\unicode[STIX]{x1D6FA})$of real-valued continuous functions on$\unicode[STIX]{x1D6FA}$, level-two large deviation principle, level-one large deviation principle, convexity properties of some maps on$\mathbb{R}^{n}$for all$n\in \mathbb{N}$. The one involving the Gateaux differentiability of$P^{\unicode[STIX]{x1D70F}}$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$\unicode[STIX]{x1D70E}$-compact subset$\unicode[STIX]{x1D6F4}$of$C(\unicode[STIX]{x1D6FA})$, (D) is equivalent to the existence of an infinite dimensional vector space$V$dense in$C(\unicode[STIX]{x1D6FA})$such that$f+g$has a unique equilibrium state for all$(f,g)\in \unicode[STIX]{x1D6F4}\times V\setminus \{0\}$; any Schauder basis$(f_{n})$of$C(\unicode[STIX]{x1D6FA})$whose linear span contains$\unicode[STIX]{x1D6F4}$admits an arbitrary small perturbation$(h_{n})$so that one can take$V=\text{span}(\{f_{n}+h_{n}:n\in \mathbb{N}\})$. Taking$\unicode[STIX]{x1D6F4}=\{0\}$, the existence of an infinite dimensional vector space dense in$C(\unicode[STIX]{x1D6FA})$constituted by functions admitting a unique equilibrium state is equivalent to (D) together with the uniqueness of the measure of maximum entropy.