Nilpotent locally compact groups with small topological entropy

Abstract

We characterize the finiteness of the topological entropy of continuous automorphisms of locally compact nilpotent p-groups (p prime) via the notion of p-rank. Considering upper unitriangular matrices over the p-adic integers and p-adic rationals, we present an algorithmic criterion in order to produce nilpotent locally compact p-groups of large nilpotency class and with continuous automorphisms of finite topological entropy. The procedure allows us to generalize the construction of large families of totally disconnected locally compact Heisenberg p-groups. It should be also mentioned that alternative arguments have been proposed, in order to avoid the use of the p-rank for the finiteness of the topological entropy of the continuous automorphisms, but these arguments involve the notion of topologically capable group, which wasn't explored for locally compact groups (except for the discrete case).