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We show that for a $\mathbb{Z}^{l}$-action (or $(\N\cup\{0\})^l$-action) on anon-empty compact metrizable space $\Omega$, the existence of a affine spacedense in the set of continuous functions on $\Omega$ constituted by elementsadmitting a unique equilibrium state implies that each invariant measure can beapproximated weakly$^*$ and in entropy by a sequence of measures which areunique equilibrium states.

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