Geometrical Models for Substitutions

Abstract

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.

2000 AMS Subject Classification:

• 37B10
• 37E05

Keywords:

• Interval exchange transformations
• Rauzy fractals
• substitutive dynamical systems
• Peano curves