Dimension and entropy in compact topological groups

We study the topological entropy h(f) of continuous endomorphisms f of compact like groups. More specifically, we consider the e-spectrum E-top(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 < infinity of topological groups without continuous endomorphisms of infinite entropy (i.e., infinity is not an element of E-top(K)) as well as the subclass E-0 of E-<infinity, consisting of those groups K with E-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 if and only if dim K < infinity 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 dim K = infinity and K is an element of 4, or countably compact connected abelian groups with the same property). Since the class E-<infinity is not stable under taking closed subgroups or quotients, we study also the largest subclasses S(E-<infinity) and Q(E-<infinity), respectively, of E-<infinity, 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(E-0) and Q(E-0) is done as well. (C) 2019 Elsevier Inc. All rights reserved.