Conjugacy of unimodular pisot substitutions subshifts to domain exchanges

TL;DR

This paper demonstrates that unimodular Pisot substitution subshifts can be represented as domain exchanges in Euclidean space, extending previous geometric realizations to all such substitutions without extra conditions.

Contribution

It proves a general conjugacy result linking unimodular Pisot substitution subshifts to domain exchanges, broadening the scope of geometric realizations.

Findings

Unimodular Pisot substitution subshifts are measurably conjugate to Euclidean domain exchanges.

This conjugacy is a finite topological extension of a torus translation.

The result generalizes previous work by removing additional combinatorial restrictions.

Abstract

We prove that any unimodular Pisot substitution subshift is measurably conjugate to a domain exchange in an Euclidean space which is a finite topological extension of a translation on a torus.This generalizes the pioneer works of Rauzy and Arnoux-Ito providing geometric realizations to any unimodular Pisot substitution without any additional combinatorial condition.