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

2016; Cambridge University Press; Volume: 37; Issue: 3 Linguagem: Inglês

10.1017/etds.2015.72

1469-4417

Henri Comman,

advanced mathematical theories

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.