Entropy functions for semigroup actions

Abstract

We consider continuous actions of finitely generated semigroups and countable sofic groups, generated either by continuous self-maps or by homeomorphisms of a compact metric space. For each known topological pressure operator associated to these actions, we provide a measure-theoretic entropy map which is concave, upper semicontinuous and satisfies a variational principle whose maximum is always attained. In the case of countable amenable group actions whose amenable entropy is concave and upper semicontinuous, we show that, for any sofic approximation sequence, the amenable metric entropy and the previous measure-theoretic entropy coincide on the space of invariant probability measures, and are equal to the upper semicontinuous envelope of the sofic entropy.