Abstract: This paper studies tilings and representation sapces related to the $\beta$-transformation when $\beta$ 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 $\beta$-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 $\beta$-expansion, called central tile: for elements $x$ of the ring $\GZ[1/\beta]$, 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 $\beta$. This representation space has numerous potential implications. We focus here 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 papers ends with explicit quadratic examples, showing that the general behaviour of $\gamma(\beta)$ is slightly more complicated than in the unit case.