Conjugacy of unimodular Pisot substitutions subshifts to domain exchanges

We prove that any unimodular Pisot substitution subshift is measurably conjugate to a domain exchange in Euclidean spaces which factorizes onto a minimal rotation 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.Comment: 29 p. In this new version, a gap in the proof of the main theorem has been fixe

Decidability of the HD0L ultimate periodicity problem

In this paper we prove the decidability of the HD0L ultimate periodicity problem

On the number of return words in infinite words with complexity 2n+1

In this article, we count the number of return words in some infinite words with complexity 2n+1. We also consider some infinite words given by codings of rotation and interval exchange transformations on k intervals. We prove that the number of return words over a given word w for these infinite words is exactly k.Comment: see also http://liafa.jussieu.fr/~vuillon/articles.htm