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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Algebraic entropy for Abelian groups


Authors: Dikran Dikranjan, Brendan Goldsmith, Luigi Salce and Paolo Zanardo
Journal: Trans. Amer. Math. Soc. 361 (2009), 3401-3434
MSC (2000): Primary 20K30; Secondary 20K10, 37A35
Published electronically: March 3, 2009
MathSciNet review: 2491886
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Abstract: The theory of endomorphism rings of algebraic structures allows, in a natural way, a systematic approach based on the notion of entropy borrowed from dynamical systems. Here we study the algebraic entropy of the endomorphisms of Abelian groups, introduced in 1965 by Adler, Konheim and McAndrew. The so-called Addition Theorem is proved; this expresses the algebraic entropy of an endomorphism $ \phi$ of a torsion group as the sum of the algebraic entropies of the restriction to a $ \phi$-invariant subgroup and of the endomorphism induced on the quotient group. Particular attention is paid to endomorphisms with zero algebraic entropy as well as to groups all of whose endomorphisms have zero algebraic entropy. The significance of this class arises from the fact that any group not in this class can be shown to have endomorphisms of infinite algebraic entropy, and we also investigate such groups. A uniqueness theorem for the algebraic entropy of endomorphisms of torsion Abelian groups is proved.


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Additional Information

Dikran Dikranjan
Affiliation: Dipartimento di Matematica e Informatica, Università di Udine, Via Delle Scienze 206, 33100 Udine, Italy
Email: dikranja@dimi.uniud.it

Brendan Goldsmith
Affiliation: School of Mathematical Sciences, Dublin Institute of Technology, Dublin 2, Ireland
Email: brendan.goldsmith@dit.ie

Luigi Salce
Affiliation: Dipartimento di Matematica Pura e Applicata, Università di Padova, Via Trieste 63, 35121 Padova, Italy
Email: salce@math.unipd.it

Paolo Zanardo
Affiliation: Dipartimento di Matematica Pura e Applicata, Università di Padova, Via Trieste 63, 35121 Padova, Italy
Email: pzanardo@math.unipd.it

DOI: https://doi.org/10.1090/S0002-9947-09-04843-0
Keywords: Algebraic entropy, endomorphism rings, Abelian groups
Received by editor(s): May 12, 2006
Published electronically: March 3, 2009
Additional Notes: The research of the first, third, and fourth authors was supported by MIUR, PRIN 2005.
Dedicated: In Memoriam: Il Maestro, Adalberto Orsatti
Article copyright: © Copyright 2009 American Mathematical Society