Homeomorphism groups of basilica, rabbit and airplane Julia sets
Bruno Duchesne, Matteo Tarocchi
TL;DR
This work analyzes the full homeomorphism groups of three well-known Julia-set fractals—the airplane and the regular rabbits (including the basilica and Douady rabbit). It builds a bridge from fractal topology to group theory by encoding the circle/cut-point structures into trees and Waжewski dendrites, then identifying the automorphism groups with universal groups and kaleidoscopic groups, realized as Polish groups. Consequences include strong structural results: simplicity of the homeomorphism groups (except for certain orientation-restricted variants), precise embedding relations among the rabbit/airplane groups, and detailed geometric/analytic properties (Roelcke precompactness, quasi-isometry types, and (T) vs Haagerup properties). The laminations associated to Julia sets are analyzed to identify lamination automorphism groups and connect them with the circle automorphism framework, yielding a unified perspective on symmetry both topological and dynamical. Finally, the paper computes the universal minimal flows for the airplane case, illustrating an explicit metrizable flow and an intrinsically cyclic-ordered structure for the oriented variant, and situates these results within the broader program of understanding rearrangement and universal groups acting on fractal spaces.
Abstract
The airplane, the basilica and the Douady rabbit (and, more generally, rabbits with more than two ears) are well-known Julia sets of complex quadratic polynomials. In this paper we study the groups of all homeomorphisms of such fractals and of all automorphisms of their laminations. In particular, we identify them with some kaleidoscopic group or universal groups and thus realize them as Polish permutation groups. From these identifications, we deduce algebraic, topological and geometric properties of these groups.