Homeomorphism groups of basilica, rabbit and airplane Julia sets

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.

💡 Research Summary

The paper investigates the full homeomorphism groups of three well‑known quadratic Julia sets: the basilica, the Douady rabbit (and its higher‑ear generalisations) and the airplane. After giving abstract topological definitions of “rabbits” (Peano continua equipped with a dense family of circles satisfying separation and finiteness conditions) and of “airplane” (a similar continuum where circles are pairwise disjoint and any two are separated by a third circle), the authors prove a uniqueness theorem: up to homeomorphism there is exactly one airplane and exactly one n‑regular rabbit for each n ≥ 2.

Each fractal carries a natural “tree of circles”: for the airplane this is the universal Ważewski dendrite D∞, for an n‑regular rabbit it is the biregular tree Tₙ,∞ (vertices of degree n alternating with vertices of infinite degree). Any homeomorphism of the fractal induces a faithful action on the associated tree or dendrite. However, not every tree automorphism lifts to a homeomorphism of the fractal because the cyclic order on each circle must be respected. To capture this, the authors introduce Aut(S), the group of all bijections of a dense countable subset of a circle that preserve the separation relation (equivalently, preserve or reverse the cyclic order).

Using the Burger–Mozes construction of universal groups U(Γ) for regular trees and Smith’s extension to biregular trees, the authors identify the homeomorphism groups as follows:

• The homeomorphism group of the airplane is the kaleidoscopic group K(Aut(S)).
• The homeomorphism group of an n‑regular rabbit is the universal group U(Sym(