Tree substitutions and Rauzy fractals

TL;DR

This paper explores the connection between tree substitutions, Rauzy fractals, and automorphisms of free groups, providing a method to approximate and analyze the associated dynamical systems and their geometric structures.

Contribution

It introduces a novel construction of tree substitutions to approximate repelling real trees and relates these to Rauzy fractals for irreducible Pisot substitutions.

Findings

Constructed tree substitutions for automorphisms of free groups.

Produced images of trees inside Rauzy fractals.

Described the contour and computed an interval exchange transformation.

Abstract

We work with attracting subshifts generated by substitutions which are also irreducible parageometric automorphisms of free groups. For such a dynamical system, we construct a tree substitution to approximate the repelling real tree of the automorphism. We produce images of this tree inside the Rauzy fractal when the substitution is irreducible Pisot. We describe the contour of this tree and compute an interval exchange transformation of the circle covering the original substitution.