Substitutions, Rauzy fractals and tilings
Valérie Berthé, Anne Siegel, Jörg Μ. Thuswaldner · Cambridge University Press eBooks · 2010
Introduction This chapter focuses on multiple tilings associated with substitutive dynamical systems. We recall that a substitutive dynamical system ( X σ , S ) is a symbolic dynamical system where the shift S acts on the set X σ 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 n–1 , 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 ↦ 12, 2 ↦ 13, 3 ↦ 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 3 – X 2 – X – 1).