Geometrical representation of subshifts for primitive substitutions

TL;DR

This paper explores the geometric and dynamical properties of subshifts generated by primitive substitutions with Pisot eigenvalues, providing constructions, conditions, and algorithms for their analysis.

Contribution

It introduces a domain exchange model for such subshifts, establishes criteria for torus translation extensions, and offers an algorithm to compute eigenvalues.

Findings

Subshifts are measurably conjugate to domain exchanges.

Conditions are given for subshifts to be finite extensions of torus translations.

An algorithm for eigenvalue computation of primitive pseudo-unimodular substitutions is provided.

Abstract

For any primitive substitution whose Perron eigenvalue is Pisot unit, we construct a domain exchange measurably conjugated to the subshift. And we give a condition for the subshift to be a finite extension of a torus translation. For the particular case of weakly irreducible Pisot substitution, we show that the subshift is either a finite extension of a torus translation, either a power of the subshift is weakly mixing. And we provide an algorithm to compute eigenvalues of the subshift associated to any primitive pseudo-unimodular substitution.