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, another Peano curve related to the work of Mc- Mullen and Lowenstein et al., and also the interval exchange transformation itself.