We study the topological entropy h(f) of continuous endomorphisms f of compact- like groups. More specifically, we consider the e-spectrum Etop(K) for a compact-like group K (namely, the set of all values h(f), when f runs over the set End(K) of all continuous endomorphisms of K). We pay particular attention to the class E<∞ of topological groups without continuous endomorphisms of infinite entropy (i.e., ∞ ∈/ Etop(K)) as well as the subclass E0 of E<∞ consisting of those groups K with Etop(K) = {0}. It turns out that the properties of the e-spectrum and these two classes are very closely related to the topological dimension. We show, among others, that a compact connected group K with finite-dimensional commutator subgroup belongs to E<∞ if and only if dim K < ∞ and we obtain a simple formula (involving the entropy function) for the dimension of an abelian topological group which is either locally compact or ω-bounded (in particular, compact). Examples are provided to show the necessity of the compactness or commutativity conditions imposed for the validity of these results (e.g., compact connected semi-simple groups K with dimK = ∞ and K ∈ E0, or countably compact connected abelian groups with the same property). Since the class E<∞ is not stable under taking closed subgroups or quotients, we study also the largest subclasses S(E<∞) and Q(E<∞), respectively, of E<∞, having these stability properties. We provide a complete description of these two classes in the case of compact groups, that are either abelian or connected. The counterpart for S(E0) and Q(E0) is done as well.

Dimension and entropy in compact topological groups

Dikranjan, Dikran;
2019-01-01

Abstract

We study the topological entropy h(f) of continuous endomorphisms f of compact- like groups. More specifically, we consider the e-spectrum Etop(K) for a compact-like group K (namely, the set of all values h(f), when f runs over the set End(K) of all continuous endomorphisms of K). We pay particular attention to the class E<∞ of topological groups without continuous endomorphisms of infinite entropy (i.e., ∞ ∈/ Etop(K)) as well as the subclass E0 of E<∞ consisting of those groups K with Etop(K) = {0}. It turns out that the properties of the e-spectrum and these two classes are very closely related to the topological dimension. We show, among others, that a compact connected group K with finite-dimensional commutator subgroup belongs to E<∞ if and only if dim K < ∞ and we obtain a simple formula (involving the entropy function) for the dimension of an abelian topological group which is either locally compact or ω-bounded (in particular, compact). Examples are provided to show the necessity of the compactness or commutativity conditions imposed for the validity of these results (e.g., compact connected semi-simple groups K with dimK = ∞ and K ∈ E0, or countably compact connected abelian groups with the same property). Since the class E<∞ is not stable under taking closed subgroups or quotients, we study also the largest subclasses S(E<∞) and Q(E<∞), respectively, of E<∞, having these stability properties. We provide a complete description of these two classes in the case of compact groups, that are either abelian or connected. The counterpart for S(E0) and Q(E0) is done as well.
File in questo prodotto:
File Dimensione Formato  
DDSanchis.pdf

Open Access dal 01/04/2022

Descrizione: Versione .pdf Editoriale
Tipologia: Versione Editoriale (PDF)
Licenza: Non pubblico
Dimensione 726.8 kB
Formato Adobe PDF
726.8 kB Adobe PDF Visualizza/Apri
CastellonC.pdf

accesso aperto

Tipologia: Documento in Pre-print
Licenza: Creative commons
Dimensione 451.78 kB
Formato Adobe PDF
451.78 kB Adobe PDF Visualizza/Apri

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11390/1148634
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 4
  • ???jsp.display-item.citation.isi??? 4
social impact