Fractal trees for irreducible automorphisms of free groups

TL;DR

This paper explores the fractal, self-similar structure of repelling trees associated with irreducible automorphisms of free groups, revealing their Hausdorff dimension and fractal properties.

Contribution

It demonstrates the self-similarity of repelling trees in Outer Space and computes their Hausdorff dimension, linking automorphism dynamics with fractal geometry.

Findings

Repelling trees exhibit self-similarity akin to fractals.

Hausdorff dimension of these trees is explicitly calculated.

Reveals fractal nature of attracting trees in Outer Space.

Abstract

The self-similar structure of the attracting subshift of a primitive substitution is carried over to the limit set of the repelling tree in the boundary of Outer Space of the corresponding irreducible outer automorphism of a free group. Thus, this repelling tree is self-similar (in the sense of graph directed constructions). Its Hausdorff dimension is computed. This reveals the fractal nature of the attracting tree in the boundary of Outer Space of an irreducible outer automorphism of a free group.