Abstract entropy and expansiveness

TL;DR

This paper introduces a unified abstract framework for entropy and expansiveness in pre-ordered semigroups, generalizing topological entropy and extending the concept of expansive systems.

Contribution

It develops a general definition of entropy in pre-ordered semigroups, unifies various entropy theories, and extends expansiveness to this abstract setting.

Findings

Basic properties of the abstract entropy are established.

The framework encompasses many existing entropy theories as special cases.

New definitions of entropy and expansiveness are proposed for specific applications.

Abstract

We present a general definition of entropy in the setting of pre-ordered semigroups, extending the notion of topological entropy. From our definition, we obtain the basic properties exhibited by various entropy-like theories encountered in the literature, many of them being particular cases of our scheme. We show how to derive from the properties of the abstract entropy corresponding properties in the concrete cases. We also introduce a notion of expansiveness in this general context, extending the concept of expansive dynamical system, which is related to the entropy in a similar fashion as in the case of topological dynamics. Finally, as an application, we suggest some definitions of new entropies and expansiveness in concrete cases.