Homeomorphism groups of basilica, rabbit and airplane Julia sets

TL;DR

This paper investigates the homeomorphism groups of specific Julia sets, identifying them with kaleidoscopic and universal groups, and explores their algebraic, topological, and geometric properties.

Contribution

It provides a novel characterization of homeomorphism groups of basilica, rabbit, and airplane Julia sets as well-known permutation groups.

Findings

Identified these groups with kaleidoscopic and universal groups

Established their algebraic, topological, and geometric properties

Realized them as Polish permutation groups

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.