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

TL;DR

This paper investigates the relationship between entropy approximation and the uniqueness of equilibrium states for dense affine spaces of continuous functions under certain group actions, revealing that unique equilibrium states can approximate all invariant measures.

Contribution

It establishes that the existence of a dense affine space with unique equilibrium states ensures all invariant measures can be approximated by such states.

Findings

Unique equilibrium states are dense in the space of invariant measures.

Invariant measures can be approximated weakly$^*$ and in entropy by unique equilibrium states.

The result applies to $ abla$-actions on compact metrizable spaces.

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.