Stepped surfaces and Rauzy fractals induced from automorphisms on the free group of rank 2

TL;DR

This paper extends the construction of Rauzy fractals and stepped surfaces from substitution systems to hyperbolic automorphisms on the free group of rank 2, revealing their geometric and dynamical properties.

Contribution

It introduces a method to derive Rauzy fractals from hyperbolic automorphisms, generalizing previous substitution-based approaches.

Findings

Set equations for Rauzy fractals are obtained.

Domain exchange on Rauzy fractals is equivalent to a two-interval exchange.

Extension of tiling methods to a broader class of automorphisms.

Abstract

For substitution satisfying Pisot, irreducible, unimodular condition, a tiling substitution plays a key role in construction of a stepped surface and Rauzy fractal. In this paper we will extend the method to hyperbolic automorphisms on the free group of rank 2 in some class, and obtain set equations of Rauzy fractals by virtue of a tiling substitution. We will also see that the domain exchange transformation on Rauzy fractal is just a two interval exchange transformation.