Following previous meetings on related subjects, we are organizing the 2006 workshop "Numeration, tilings and substitutions" in Marseille, March 6--10 2006, in the former CNRS learning center, CIRM, in the large annex room. The goal of this informal workshop is to gather reasearchers around beta-numerations in algebraic bases (Pisot, Salem) and generalized substitutions, with special focus on the connections with tilings and with the Pisot conjecture: It is possible, generalizing Rauzy's and Thurston's constructions, to associate in a natural way either with a Pisot number beta (of degree d) or with a Pisot substitution (on d letters) some compact basic tiles that are the closure of their interior, that have non-zero measure and a fractal boundary; they are attractors of some graph-directed Iterated Function System, and are called Rauzy fractals (other examples: ex1, ex2), or central tiles. We know that some translates of these prototiles under a Delone set (provided by beta or by the substitution) cover the d-1-dimensional space; it is conjectured that this multiple tiling is indeed a tiling (which might be either periodic or self-replicating according to the translation set). This conjecture is known as the Pisot conjecture and can also be reformulated in spectral terms: the associated dynamical systems have pure discrete spectrum. The topics involved thus cover a large panel of common tools developed in number theory, combinatorics, algorithmics, symbolic dynamics, and geometry.
This workshop, mainly supported by
The idea of this workshop is to devote a maximum of time to "small group discussions" instead of plenary talks. Hence, we propose the following:
Information on the acces to CIRM is provided here.
Below is the program and some presentations:
Some photos taken during the workshop.
Below is the list of participants.