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

Abstract

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