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 $(\N\cup\{0\})^l$-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$^*$ and in entropy by a sequence of measures which are unique equilibrium states.