Algebraic expansivity on abelian groups

TL;DR

This paper introduces algebraic expansivity for endomorphisms of abelian groups, linking it to algebraic entropy and topological dynamics, and characterizes positively expansive epimorphisms as finite systems.

Contribution

It defines algebraic expansivity, explores its properties, and establishes its duality with topological expansivity via Pontryagin duality.

Findings

Positively expansive epimorphisms are finite systems.

Algebraic expansivity relates to Weiss's algebraic entropy.

Duality connects algebraic and topological expansivity on torsion groups.

Abstract

Building on the author's earlier work on topological and abstract expansivity, this paper introduces and explores the notion of algebraic expansivity for endomorphisms of abelian groups. We analyze the fundamental properties of this algebraic analogue, establish its relationship with Weiss's algebraic entropy, and prove that positively expansive epimorphisms are necessarily restricted to finite systems. Finally, we demonstrate a robust connection with topological dynamics via Pontryagin duality: algebraic expansivity on torsion abelian groups is shown to be exactly the dual property of topological expansivity on totally disconnected compact groups.