Quasisymmetries of the basilica and the Thompson group

TL;DR

This paper characterizes the group of all quasisymmetric self-maps of the Julia set of a quadratic polynomial, showing it is generated by the Thompson group and an inversion, with controlled distortions.

Contribution

It provides a precise description of the quasisymmetric automorphism group of the Julia set, linking it to the Thompson group and inversion, with quantitative distortion control.

Findings

The group of quasisymmetric self-maps is the uniform closure of the generated group.

Distortions of approximating maps are uniformly controlled.

The result connects complex dynamics with geometric group theory.

Abstract

We give a description of the group of all quasisymmetric self-maps of the Julia set of $f(z)=z^2-1$ that have orientation preserving homeomorphic extensions to the whole plane. More precisely, we prove that this group is the uniform closure of the group generated by the Thompson group of the unit circle and an inversion. Moreover, this result is quantitative in the sense that distortions of the approximating maps are uniformly controlled by the distortion of the given map.