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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 $(\N\cup\{0\})^l$-action) $τ$ on a non-empty compact metrisable space $Ω$, for some $l\in\N$. Let (D) denote the following property: The graph of the restriction of the entropy map $h^τ$ to the set of ergodic states is dense in the graph of $h^τ$. We assume that $h^τ$ 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 $P^τ$ on some sets dense in the space $C(Ω)$ of real-valued continuous functions on $Ω$, 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 $P^τ$ 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 $σ$-compact subset $Σ$ of $C(Ω)$, (D) is equivalent to the existence of an infinite dimensional vector space $V$ dense in $C(Ω)$ such that $f+g$ has a unique equilibrium state for all $(f,g)\in Σ\times V\setminus\{0\}$; any Schauder basis $(f_n)$ of $C(Ω)$ whose linear span contains $Σ$ admits an arbitrary small perturbation $(h_n)$ so that one can take $V=\textnormal{span}(\{f_n+h_n: n\in\N\})$. Taking $Σ=\{0\}$, the existence of an infinite dimensional vector space dense in $C(Ω)$ constituted by functions admitting a unique equilibrium state is equivalent to (D) together with the uniqueness of measure of maximal entropy.