Quasisymmetries of the basilica and the Thompson group
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.