Boundary of central tiles associated with Pisot beta-numeration and purely periodic expansions

This paper studies tilings and representation sapces related to the β-transformation when β is a Pisot number (that is not supposed to be a unit). The obtained results are applied to study the set of rational numbers having a purely periodic β-expansion. We indeed make use of the connection between pure periodicity and a compact self-similar representation of numbers having no fractional part in their β-expansion, called central tile: for elements x of the ring $${\mathbb {Z}[1/\beta]}$$ , so-called x-tiles are introduced, so that the central tile is a finite union of x-tiles up to translation. These x-tiles provide a covering (and even in some cases a tiling) of the space we are working in. This space, called complete representation space, is based on Archimedean as well as on the non-Archimedean completions of the number field $${{\mathbb Q} (\beta)}$$ corresponding to the prime divisors of the norm of β. This representation space has numerous potential implications. We focus here on the gamma function γ(β) defined as the supremum of the set of elements v in [0, 1] such that every positive rational number p/q, with p/q  ≤  v and q coprime with the norm of β, has a purely periodic β-expansion. The key point relies on the description of the boundary of the tiles in terms of paths on a graph called “boundary graph”. The paper ends with explicit quadratic examples, showing that the general behaviour of γ(β) is slightly more complicated than in the unit case.