Abstract Numeration Systems and Tilings
An abstract numeration system is a triple S = (L,Σ,<) where (Σ,<) is a totally ordered alphabet and L a regular language over Σ; the associated numeration is defined as follows: by enumerating the words of the regular language L over Σ 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 ăthe 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 , and for central tiles associated with a Pisot beta-numeration  .
Unable to display preview. Download preview PDF.
- 1.Akiyama, S.: Self affine tiling and Pisot numeration system. In: Number theory and its applications (Kyoto 1997). Dev. Math, vol. 2, pp. 7–17. Kluwer Acad. Publ, Dordrecht (1999)Google Scholar
- 3.Berthé, V., Rigo, M.: Odometers on regular languages. Theory Comput. Syst (to appear)Google Scholar
- 4.Berthé, V., Siegel, A.: Purely periodic expansions in the non-unit case (2004) (preprint)Google Scholar
- 5.Berthé, V., Siegel, A.: Tilings associated with beta-numeration and substitutions. Integers: Electronic Journal of Combinatorial Number Theory (to appear)Google Scholar
- 22.Thomas, W.: Automata on infinite objects. In: Handbook of theoret. comput. sci., vol. B, pp. 133–191. Elsevier, Amsterdam (1990)Google Scholar
- 23.Thurston, W.P.: Groups, tilings and finite state automata, Lectures notes distributed in conjunction with the Colloquium Series. AMS Colloquium lectures (1989)Google Scholar