Adjoint algebraic entropy

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.

27 pages

References in corpus (1)

• Adjoint entropy vs Topological entropy

Cited by in corpus (8)

• Entropy on abelian groups
• Limit free computation of entropy
• Topological entropy for locally linearly compact vector spaces
• Strings of group endomorphisms
• Topological entropy for automorphisms of totally disconnected locally compact groups
• Adjoint entropy vs Topological entropy
• Entropy in a category
• Functorial topologies and finite-index subgroups of abelian groups