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FINITENESS OF TOPOLOGICAL ENTROPY FOR LOCALLY COMPACT ABELIAN GROUPS

Part of: Topological and differentiable algebraic systems Topological dynamics Maps and general types of spaces defined by maps

Published online by Cambridge University Press:  26 February 2020

DIKRAN DIKRANJAN Open the ORCID record for DIKRAN DIKRANJAN [Opens in a new window] ,
ANNA GIORDANO BRUNO Open the ORCID record for ANNA GIORDANO BRUNO [Opens in a new window]  and
FRANCESCO G. RUSSO Open the ORCID record for FRANCESCO G. RUSSO [Opens in a new window]
DIKRAN DIKRANJAN
Affiliation:
Dipartimento di Scienze Matematiche, Informatiche e Fisiche, Università di Udine, Via delle Scienze 206, 33100Udine, Italy e-mails: [email protected]; [email protected]
ANNA GIORDANO BRUNO
Affiliation:
Dipartimento di Scienze Matematiche, Informatiche e Fisiche, Università di Udine, Via delle Scienze 206, 33100Udine, Italy e-mails: [email protected]; [email protected]
FRANCESCO G. RUSSO
Affiliation:
Department of Mathematics and Applied Mathematics, University of Cape Town, Private Bag X1, Rondebosch 7701, Cape Town, South Africa e-mail: [email protected]
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Abstract

We study the locally compact abelian groups in the class ${\mathfrak E_{ \lt \infty }}$ , that is, having only continuous endomorphisms of finite topological entropy, and in its subclass $\mathfrak E_0$ , that is, having all continuous endomorphisms with vanishing topological entropy. We discuss the reduction of the problem to the case of periodic locally compact abelian groups, and then to locally compact abelian p-groups. We show that locally compact abelian p-groups of finite rank belong to ${\mathfrak E_{ \lt \infty }}$ , and that those of them that belong to $\mathfrak E_0$ are precisely the ones with discrete maximal divisible subgroup. Furthermore, the topological entropy of endomorphisms of locally compact abelian p-groups of finite rank coincides with the logarithm of their scale. The backbone of the paper is the Addition Theorem for continuous endomorphisms of locally compact abelian groups. Various versions of the Addition Theorem are established in the paper and used in the proofs of the main results, but its validity in the general case remains an open problem.

MSC classification

Primary: 22A05: Structure of general topological groups 37B40: Topological entropy
Secondary: 54C70: Entropy
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 63 , Issue 1 , January 2021 , pp. 81 - 105
Copyright
© The Author(s) 2020. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

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