From some Pisot numerations to topological groups

Abstract

A Pisot numeration system $U$ for $\mathbb{N}$ is a sequence of natural numbers generated by an integral homogeneous linear recurrence whose characteristic polynomial is the minimal polynomial of a Pisot number. The purpose of this paper is to introduce the analogue of the group of $p$-adic integers for such numerations when they \emph{preserve zeros}, which is equivalent to the `Condition F' introduced by Frougny and Solomyak for $\beta$-numerations. We show that these topological groups ${\mathbb{Z}}_U$ project homomorphically onto a torus. Equipping ${\mathbb{Z}}_U$ with the appropriate topology, we also show that if $U$ is unimodular, then ${\mathbb{Z}}_U$ is continuously isomorphic to a torus.

29 pages, 4 figures. Minor changes in version 2