Delone sets of beta-integers, Rauzy fractals and adeles

摘要

We recall the two canonical cut-and-project schemes which lie over the set Z_β of β-integers when β is a Perron number of degree m ≥ 2. If β is Pisot only, the canonical windows in the two internal spaces are fractals, called Rauzy fractal in one of them, and Z_β is a Meyer set. The existence of supplementary p-adic internal spaces with p dividing the norm N(β), shown recently by A. Siegel in the study of tile dynamics of substitution systems of Pisot type, in which Rauzy fractals are extended, suggests to consider cut-and-project schemes over the ring of adeles of Q(β) as a complete geometrical framework over Z_β. This indicates a deep link with the works of A. Weil on adeles, already partially discovered by J.-P. Schreiber.