Topological Entropy and Algebraic Entropy for group endomorphisms

TL;DR

This survey explores topological and algebraic entropy in the context of group endomorphisms, introducing new entropy functions, generalizations, and connections to number theory and geometric group theory.

Contribution

It provides a comprehensive overview of entropy in group endomorphisms, introduces new entropy functions, and links these concepts to other mathematical areas.

Findings

Introduction of new entropy functions

Generalizations of existing entropy definitions

Connections to Mahler measure, Lehmer Problem, and group growth rates

Abstract

The notion of entropy appears in many fields and this paper is a survey about entropies in several branches of Mathematics. We are mainly concerned with the topological and the algebraic entropy in the context of continuous endomorphisms of locally compact groups, paying special attention to the case of compact and discrete groups respectively. The basic properties of these entropies, as well as many examples, are recalled. Also new entropy functions are proposed, as well as generalizations of several known definitions and results. Furthermore we give some connections with other topics in Mathematics as Mahler measure and Lehmer Problem from Number Theory, and the growth rate of groups and Milnor Problem from Geometric Group Theory. Most of the results are covered by complete proofs or references to appropriate sources.