Homeomorphism groups of basilica, rabbit and airplane Julia sets
Dynamical Systems
2026-02-03 v2 General Topology
Group Theory
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.
Cite
@article{arxiv.2502.07762,
title = {Homeomorphism groups of basilica, rabbit and airplane Julia sets},
author = {Bruno Duchesne and Matteo Tarocchi},
journal= {arXiv preprint arXiv:2502.07762},
year = {2026}
}