On locally compact groups of small topological entropy

Abstract

We discuss the finiteness of the topological entropy of continuous endomorphims for some classes of locally compact groups. Firstly, we focus on the abelian case, imposing the condition of being compactly generated, and note an interesting behaviour of slender groups. Secondly, we remove the condition of being abelian and consider nilpotent periodic locally compact $p$-groups ($p$ prime), reducing the computations to the case of Sylow $p$-subgroups. Finally, we investigate locally compact Heisenberg $p$-groups $\mathbb{H}_{n}(\mathbb{Q}_{p})$ on the field $\mathbb{Q}_{p}$ of the $p$-adic rationals with $n$ arbitrary positive integer.

Theorems & Definitions (31)

• Theorem 1.1: See, hofmor, Theorem 7.57
• Definition 1.2: See Fuchs, p.489
• Theorem 1.3: First Main Theorem
• Theorem 1.4: Second Main Theorem
• Theorem 1.5: Third Main Theorem
• Lemma 2.1: See Fuchs, Chapter 13, § 2
• Lemma 2.2: See hofmor, Corollary 8.5
• Lemma 2.3: See Fuchs, Lemma 2.3, Sasiada's Theorem
• Definition 2.4: See DeMarcoOrsatti1974
• Theorem 2.5: See DeMarcoOrsatti1974, Theorem 2.3