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Cet article est consacre a l'etude du flot de translation sur pavages auto-similaires associes a une substitution de type Pisot. Nous construisons une representation geometrique et nous donnons les conditions necessaires et suffisantes pour que le flot ait un spectre purement discret. Dans l'application, nous montrons que pour certains beta-shifts, l'extension naturelle est naturellement isomorphique a un automorphisme du tore.

Geometric realization and coincidence for reducible non-unimodular Pisot tiling spaces with an application to \beta -shifts