Geometrical representation of subshifts for primitive substitutions

Paul Mercat

Published 2022-01-04, updated 2023-09-20Version 2

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.