Adjoint algebraic entropy

TL;DR

This paper introduces the concept of adjoint algebraic entropy for endomorphisms of Abelian groups, explores its properties, and establishes key results including a dichotomy and a discontinuity criterion.

Contribution

It defines adjoint algebraic entropy, proves its relation to algebraic entropy via Pontryagin duality, and establishes fundamental theorems including an Addition Theorem and a dichotomy.

Findings

Adjoint algebraic entropy equals algebraic entropy of the dual endomorphism.

The entropy of any endomorphism is either zero or infinity.

Finite positive algebraic entropy implies discontinuity of the endomorphism.

Abstract

The new notion of adjoint algebraic entropy of endomorphisms of Abelian groups is introduced. Various examples and basic properties are provided. It is proved that the adjoint algebraic entropy of an endomorphism equals the algebraic entropy of the adjoint endomorphism of the Pontryagin dual. As applications, we compute the adjoint algebraic entropy of the shift endomorphisms of direct sums, and we prove an Addition Theorem for the adjoint algebraic entropy of bounded Abelian groups. A dichotomy is established, stating that the adjoint algebraic entropy of any endomorphism can take only values zero or infinity. As a consequence, we obtain the following surprising discontinuity criterion for endomorphisms: every endomorphism of a compact abelian group, having finite positive algebraic entropy, is discontinuous.