Adjoint algebraic entropy

Details Adjoint algebraic entropy Adjoint algebraic entropy

2010-06-27

arxiv.org/abs/1006.5204v1

Journal of Algebra 324 (2010), 442-463

Mathematics

Group Theory

27 pages

Scientific paper

10.1016/j.jalgebra.2010.03.025

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.

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