Entropy of a semigroup of maps from a set-valued view

TL;DR

This paper introduces Hausdorff metric entropy as a new invariant for finitely generated semigroups acting on compact metric spaces, exploring its properties, relations to topological entropy, and implications for chaos theory.

Contribution

It defines and studies Hausdorff metric entropy, linking it to existing topological entropy and extending chaos notions to set-valued semigroup actions.

Findings

Hausdorff metric entropy is related to topological entropy of semigroups.

Examples demonstrate cases with positive or zero Hausdorff metric entropy.

Generalizations of chaos notions for set-valued semigroup actions are provided.

Abstract

In this paper, we introduce a new entropy-like invariant, named Hausdorff metric entropy, for finitely generated semigroups acting on compact metric spaces from a set-valued view and study its properties. We establish the relation between Hausdorff metric entropy and topological entropy of a semigroup defined by Bi\'s. Some examples with positive or zero Hausdorff metric entropy are given. Moreover, some notions of chaos are also well generalized for finitely generated semigroups from a set-valued view.