This paper under review studies tilings and representation spaces related to the \(\beta\)-transformation when \(\beta\) is a Pisot number, that is not a unit. The obtained results are applied to study the set of rational numbers having a purely periodic \(\beta\)-expansion. The authors make use of the connection between pure periodicity and a compact self-similar representation of number having no fractional part in their \(\beta\)-expansion, called center 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 the authors working in. This space, called complete representation space, is based on archimedean as well as on the non-archimedean completions of the number field \(\mathbb Q(b)\) corresponding to the prime divisors of the norm of \(\beta\). This representation space has numerous potential implications. The authors focus on the gamma function \(\gamma(\beta)\) defined as the supremum of the set of elements \(v\) in \([0,1]\) such that every positive rational number \(p/q\), with \(p/q\leq v\) and \(q\) coprime with the norm of \(\beta\), has a purely periodic \(\beta\)-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 \(\gamma(\beta)\) is slightly more complicated than in the unit case.