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  • Dimension and entropy in compact topological groups
    Dikranjan, Dikran N., 1950- ; Sanchis, Manuel
    We study the topological entropy ▫$h(f)$▫ of continuous endomorphisms ▫$f$▫ of compact-like groups. More specifically, we consider the e-spectrum ▫$\boldsymbol{E}_{\text{top}} (K)$▫ for a ... compact-like group ▫$K$▫ (namely, the set of all values ▫$h(f)$▫, when ▫$f$▫ runs over the set ▫$\text{End}(K))$▫ of all continuous endomorphisms of ▫$K$▫). We pay particular attention to the class ▫$\mathfrak{E}_{< \infty}$▫ of topological groups without continuous endomorphisms of infinite entropy (i.e., ▫$\infty \notin \boldsymbol{E}_{\text{top}} (K)$▫) as well as the subclass ▫$\mathfrak{E}_0$▫ of ▫$\mathfrak{E}_{<\infty}$▫ consisting of those groups ▫$K$▫ with ▫$\boldsymbol{E}_{\text{top}} (K) = \{0\}$▫. 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 ▫$\mathfrak{E}_{<\infty}$▫ if and only if ▫$\dim K < \infty$▫ and we obtain a simple formula (involving the entropy function) for the dimension of an abelian topological group which is either locally compact or ▫$\omega$▫-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 ▫$K = \infty$▫ and ▫$K \in \mathfrak{E}_0$▫, or countably compact connected abelian groups with the same property). Since the class ▫$\mathfrak{E}_{< \infty}$▫ is not stable under taking closed subgroups or quotients, we study also the largest subclasses ▫$S(\mathfrak{E}_{< \infty})$▫ and ▫$Q(\mathfrak{E}_{< \infty})$▫, respectively, of ▫$\mathfrak{E}_{< \infty}$▫, 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 ▫$S(\mathfrak{E}_{0})$▫ and ▫$Q(\mathfrak{E}_{0})$▫ is done as well.
    Vir: Journal of mathematical analysis and applications. - ISSN 0022-247X (Vol. 476, iss. 2, Aug. 2019, str. 337-366)
    Vrsta gradiva - članek, sestavni del ; neleposlovje za odrasle
    Leto - 2019
    Jezik - angleški
    COBISS.SI-ID - 18663769

    Povezava(-e):

    https://doi.org/10.1016/j.jmaa.2019.03.042
    DOI

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