Geometrical Models for Substitutions

Pierre Arnoux (1) , Julien Bernat (2) , Xavier Bressaud (3)

1 IML - Institut de mathématiques de Luminy
2 IECN - Institut Élie Cartan de Nancy
3 IMT - Institut de Mathématiques de Toulouse UMR5219

We consider a substitution associated with the Arnoux-Yoccoz interval exchange transformation (IET) related to the tribonacci substitution. We construct the so-called stepped lines associated with the fixed points of the substitution in the abelianization (symbolic) space. We analyze various projections of the stepped line, recovering the Rauzy fractal, a Peano curve related to work in [Arnoux 88], another Peano curve related to the work of [McMullen 09] and [Lowenstein et al. 07], and also the interval exchange transformation itself.

Mots clés: substitutive dynamical systems; Rauzy fractals; Interval exchange transformations; Peano curves

Domaines: Systèmes dynamiques [math.DS]

Pierre Arnoux, Julien Bernat, Xavier Bressaud. Geometrical Models for Substitutions. Experimental Mathematics, 2011, 20 (1), pp.97-127. ⟨10.1080/10586458.2011.544590⟩. ⟨hal-00965169⟩