Bowen’s Entropy for Endomorphisms of Totally Bounded Abelian Groups

  • Domingo Alcaraz
  • Dikran Dikranjan
  • Manuel SanchisEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 80)


We say that the completion theorem holds for a uniform space \((X,\fancyscript{U})\) if, for every uniformly continuous function \(\alpha :(X,\fancyscript{U})\rightarrow (X,\fancyscript{U})\), the Bowen entropy of \(\alpha \) coincides with the Bowen entropy of \(\widetilde{\alpha }\), the extension of \(\alpha \) to the completion \((\widetilde{X},\widetilde{\fancyscript{U}})\) of \((X,\fancyscript{U})\). We study the completion theorem in the realm of abelian topological groups. Namely, we prove that it fails to be true in a drastic way by showing that every (abstract) abelian group \(G\) can be endowed with a totally bounded group topology \(\tau \) such that the topological group \((G,\tau )\) has endomorphisms of zero entropy whose extension to the Raĭkov completion of \((G,\tau )\) has infinite entropy. Our proof uses the structure theorems for abelian groups, the properties of the Bohr topology and Pontryagin duality. The case of metrizable groups is also analyzed in the case of Bernoulli shifts of finite groups, dense subgroups of the circle \(\mathbb T\) and the reals \(\mathbb R\). The key tool is the so-called \(e\)-supporting family of an endomorphism \(\alpha \) of an abelian metrizable group \(G\) with respect to a neighborhood \(U\) of the neutral element of \(G\). Several open questions are proposed.


Bowen’s entropy Topological entropy Bohr compactification Totally bounded group Metrizable group 



The authors wish to thank the Editors for their kind invitation to participate in this Special Issue. Our contribution is dedicated to Prof. J. Kakol on occasion of his 60th birthday. The first authors was partially supported by “Progetti di Eccellenza 2011/12" of Fondazione CARIPARO. The third author was supported by the Spanish Ministry of Science and Education (Grant number MTM2011-23118), and by Bancaixa (Projecte P1-1B2011-30).


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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Domingo Alcaraz
    • 1
  • Dikran Dikranjan
    • 2
  • Manuel Sanchis
    • 3
    Email author
  1. 1.Departamento de Matemática Aplicada y EstadísticaUniversidad Politécnica de CartagenaCartagenaSpain
  2. 2.Dipartimento de Matematica e InformaticaUniversità di UdineUdineItaly
  3. 3.Institut de Matemàtiques i Aplicacions de Castelló (IMAC)Universitat Jaume I de CastellóCastellóSpain

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