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

We consider a non-uniquely ergodic dynamical system given by a ${\mathbb{Z}}^l$-action (or $(\mathbb{N}\cup \{0\}{)}^l$-action) $\tau$ on a non-empty compact metrisable space $\mathrm{\Omega }$, for some $l\in \mathbb{N}$. 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 $C(\mathrm{\Omega })$ of real-valued continuous functions on $\mathrm{\Omega }$, level-2 large deviation principle, level-1 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 $P^{\tau }$ 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 $\mathrm{\Sigma }$ of $C(\mathrm{\Omega })$, (D) is equivalent to the existence of an infinite dimensional vector space $V$ dense in $C(\mathrm{\Omega })$ such that $f+g$ has a unique equilibrium state for all $(f,g)\in \Sigma\times V\setminus{0}$;anySchauderbasis$(f_n)$of$C(\Omega)$ whose linear span contains $\mathrm{\Sigma }$ admits an arbitrary small perturbation $(h_n)$ so that one can take $V=\text{span}(\{f_n+h_n:n\in \mathbb{N}\})$. Taking $\mathrm{\Sigma }=\{0\}$, the existence of an infinite dimensional vector space dense in $C(\mathrm{\Omega })$ constituted by functions admitting a unique equilibrium state is equivalent to (D) together with the uniqueness of measure of maximal entropy.