Quasisymmetries of the basilica and the Thompson group

2016/12/27 by Mikhail Lyubich, Lyubich, Mikhail, Sergei Merenkov +1 · 2 citations
Mathematics · #Analytic and geometric function theory #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Group Theory (math.GR) #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.1612.08492

openalex publication_date 2016/12/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We give a description of the group of all quasisymmetric self-maps of the Julia set of f(z)=z²-1 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.

Citations

• The boundary correspondence under quasiconformal mappings

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