On the measure-theoretic entropy and topological pressure of free semigroup actions(to appear in ETDS)

TL;DR

This paper extends the concepts of measure-theoretic entropy and topological pressure to free semigroup actions on compact metric spaces, establishing key relationships and principles, and applying them to affine transformations.

Contribution

It introduces new definitions for entropy and pressure in the context of free semigroup actions and proves fundamental properties and a partial variational principle.

Findings

Established the relationship between skew-product topological pressure and free semigroup pressure.

Proved a partial variational principle for topological pressure.

Applied the results to affine transformations on metrizable groups.

Abstract

In this paper, we introduce the notions of topological pressure and measure-theoretic entropy for a free semigroup action. Suppose that a free semigroup acts on a compact metric space by continuous self-maps. To this action, we assign a skew-product transformation whose fiber topological pressure is taken to be the topological pressure of the initial action. Some properties of these two notions are given, and then we give two main results. One is the relationship between the topological pressure of the skew-product transformation and the topological pressure of the free semigroup action, the other is the partial variational principle about the topological pressure. Moreover, we apply this partial variational principle to study the measure-theoretic entropy and the topological entropy of finite affine transformations on a metrizable group.