This article is devoted to the study of the translation flow on self-similar tilings associated with a substitution of Pisot type. We construct a geometric representation and give necessary and sufficient conditions for the flow to have pure discrete spectrum. As an application we demonstrate that, for certain beta-shifts, the natural extension is naturally isomorphic to a toral automorphism.

We investigate in a geometrical way the point sets of  $ℝ$  obtained by the  $\beta$-numeration that are the  $\beta$-integers  ${\mathbb{Z}}_{\beta }\subset \mathbb{Z}\left[\beta \right]$  where  $\beta$  is a Perron number. We show that there exist two canonical cut-and-project schemes associated with the  $\beta$-numeration, allowing to lift up the  $\beta$-integers to some points of the lattice  ${\mathbb{Z}}^m$  ($m=$  degree of  $\beta$) lying about the dominant eigenspace of the companion matrix of  $\beta$ . When  $\beta$  is in particular a Pisot number, this framework gives another proof of the fact that  ${\mathbb{Z}}_{\beta }$  is...