The geometry of non-unit Pisot substitutions

TL;DR

This paper explores the geometric and topological properties of Rauzy fractals linked to non-unit Pisot substitutions, connecting them with numeration systems, model sets, and dynamical systems.

Contribution

It introduces multiple approaches to defining Rauzy fractals for non-unit Pisot substitutions and studies their properties and relations to various mathematical structures.

Findings

Rauzy fractals are shown to have specific tiling properties.

Connections established between Rauzy fractals and subshifts, adic transformations, and domain exchanges.

Examples illustrate the theoretical concepts for two and three letter substitutions.

Abstract

Let $\sigma$ be a non-unit Pisot substitution and let $\alpha$ be the associated Pisot number. It is known that one can associate certain fractal tiles, so-called \emph{Rauzy fractals}, with $\sigma$. In our setting, these fractals are subsets of a certain open subring of the ad\`ele ring $\mathbb{A}_{\mathbb{Q}(\alpha)}$. We present several approaches on how to define Rauzy fractals and discuss the relations between them. In particular, we consider Rauzy fractals as the natural geometric objects of certain numeration systems, define them in terms of the one-dimensional realization of $\sigma$ and its dual (in the spirit of Arnoux and Ito), and view them as the dual of multi-component model sets for particular cut and project schemes. We also define stepped surfaces suited for non-unit Pisot substitutions. We provide basic topological and geometric properties of Rauzy fractals associated with non-unit Pisot substitutions, prove some tiling results for them, and provide relations to subshifts defined in terms of the periodic points of $\sigma$, to adic transformations, and a domain exchange. We illustrate our results by examples on two and three letter substitutions.