Entropy approximation versus uniqueness of equilibrium for a dense affine space of continuous functions

We show that for a $\mathbb{Z}^{l}$-action (or $(\N\cup\{0\})^l$-action) on a non-empty compact metrizable space $Ω$, the existence of a affine space dense in the set of continuous functions on $Ω$ constituted by elements admitting a unique equilibrium state implies that each invariant measure can be approximated weakly$^*$ and in entropy by a sequence of measures which are unique equilibrium states.

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