## Institute for Mathematical Sciences

## Preprint ims16-04

** Mikhail Lyubich and Sergei Merenkov**
*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.

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