Substitutions, Rauzy fractals and tilings

This chapter focuses on multiple tilings associated with substitutive dynamical systems. We recall that a substitutive dynamical system (Xσ, S) is a symbolic dynamical system where the shift S acts on the set Xσ of infinite words having the same language as a given infinite word which is generated by powers of a primitive substitution σ. We restrict to the case where the inflation factor of the substitution σ is a unit Pisot number. With such a substitution σ, we associate a multiple tiling composed of tiles which are given by the unique solution of a set equation expressed in terms of a graph associated with the substitution σ: these tiles are attractors of a graph-directed iterated function system (GIFS). They live in R, where n stands for the cardinality of the alphabet of the substitution. Each of these tiles is compact, it is the closure of its interior, it has non-zero measure and it has a fractal boundary that is also an attractor of a GIFS. These tiles are called central tiles or Rauzy fractals, according to G. Rauzy who introduced them in (Rauzy 1982). Central tiles were first introduced in (Rauzy 1982) for the case of the Tribonacci substitution (1 7→ 12, 2 7→ 13, 3 7→ 1), and then in (Thurston 1989) for the case of the beta-numeration associated with the Tribonacci number (which is the positive root of X − X − X − 1). One motivation for Rauzy’s construction was to exhibit explicit factors of the substitutive dynamical system (Xσ, S) as translations on compact abelian groups, under the hypothesis that σ is a Pisot substitution. By extending the seminal construction in (Rauzy 1982), it has been proved that central tiles can be associated with Pisot substitutions (see for instance (Arnoux and Ito 2001) or (Canterini and Siegel 2001b)) as well as