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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-emptycompact metrisable space $\Omega$, for some $l\in\N$. Let (D) denote thefollowing property: The graph of the restriction of the entropy map $h^\tau$ tothe 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 orderthat (D) holds, each of which is stated in terms of a basic notion: Gateauxdifferentiability of the pressure map $P^\tau$ on some sets dense in the space$C(\Omega)$ of real-valued continuous functions on $\Omega$, level-2 largedeviation principle, level-1 large deviation principle, convexity properties ofsome maps on $\R^n$ for all $n\in\N$. The one involving the Gateauxdifferentiability of $P^\tau$ is of particular relevance in the context oflarge deviations since it establishes a clear comparison with anotherwell-known sufficient condition: We show that for each non-empty$\sigma$-compact subset $\Sigma$ of $C(\Omega)$, (D) is equivalent to theexistence of an infinite dimensional vector space $V$ dense in $C(\Omega)$ suchthat $f+g$ has a unique equilibrium state for all $(f,g)\in \Sigma\timesV\setminus\{0\}$; any Schauder basis $(f_n)$ of $C(\Omega)$ whose linear spancontains $\Sigma$ admits an arbitrary small perturbation $(h_n)$ so that onecan take $V=\textnormal{span}(\{f_n+h_n: n\in\N\})$. Taking $\Sigma=\{0\}$, theexistence 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.

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