Abstract numeration systems and tilings

An abstract numeration system is a triple S = (L, Sigma, <) where (Z, <) is a totally ordered alphabet and L a regular language over Z; the associated numeration is defined as follows: by enumerating the words of the regular language L over Z with respect to the induced genealogical ordering, one obtains a one-to-one correspondence between N and L. Furthermore, when the language L is assumed to be exponential, real numbers can also be expanded. The aim of the present paper is to associate with S a self-replicating multiple tiling of athe space, under the following assumption: the adjacency matrix of the trimmed minimal automaton recognizing L is primitive with a dominant eigenvalue being a Pisot unit. This construction generalizes the classical constructions performed for Rauzy fractals associated with Pisot substitutions [16], and for central tiles associated with a Pisot beta-numeration [23].