Elsevier

Journal of Algebra

Volume 324, Issue 3, 1 August 2010, Pages 442-463
Journal of Algebra

Adjoint algebraic entropy

https://doi.org/10.1016/j.jalgebra.2010.03.025Get rights and content
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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 the 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. This resolves in a strong way an open question from [7].

MSC

primary
20K30
secondary
28D20
22D35

Keywords

Algebraic entropy
Adjoint algebraic entropy
Pontryagin duality
Endomorphisms rings
Abelian groups

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