Adjoint entropy vs Topological entropy

TL;DR

This paper introduces a topological version of adjoint entropy for continuous endomorphisms of topological abelian groups, compares it with topological entropy, and establishes connections via duality theorems.

Contribution

It generalizes adjoint algebraic entropy to topological groups, defines topological adjoint entropy, and proves Bridge Theorems linking it with algebraic entropy.

Findings

Topological adjoint entropy equals topological entropy on totally disconnected compact abelian groups.

The paper establishes duality-based Bridge Theorems connecting topological and algebraic entropy.

The new entropy measure provides a framework for comparing different entropy notions in topological groups.

Abstract

Recently the adjoint algebraic entropy of endomorphisms of abelian groups was introduced and studied. We generalize the notion of adjoint entropy to continuous endomorphisms of topological abelian groups. Indeed, the adjoint algebraic entropy is defined using the family of all finite-index subgroups, while we take only the subfamily of all open finite-index subgroups to define the topological adjoint entropy. This allows us to compare the (topological) adjoint entropy with the known topological entropy of continuous endomorphisms of compact abelian groups. In particular, the topological adjoint entropy and the topological entropy coincide on continuous endomorphisms of totally disconnected compact abelian groups. Moreover, we prove two Bridge Theorems between the topological adjoint entropy and the algebraic entropy using respectively the Pontryagin duality and the precompact duality.