Homeomorphism groups of basilica, rabbit and airplane Julia sets

The paper proves precisely the two identifications the model claims: Homeo(A) ≅ K(Aut(S)) (Theorem 2.16) and, for every n ≥ 2, Homeo(Rn) ≅ U(Sym([n]), Aut(S)) (Theorem 1.29). The proofs proceed by (i) collapsing the circles of the airplane to obtain the Ważewski dendrite D∞, choosing a kaleidoscopic coloring compatible with circle orders, and showing the natural homomorphism Π is a bicontinuous isomorphism; and (ii) encoding an n-regular rabbit by its (n,∞)-biregular tree of circles with a legal coloring so that local actions match Aut(S) at circle-vertices and Sym([n]) at cut points, again showing the induced Π is a bicontinuous isomorphism. These steps are established in Proposition 2.5, Lemma 2.9, Propositions 2.13–2.15 (culminating in Theorem 2.16) for the airplane, and in Lemma 1.21, Proposition 1.26, and Theorem 1.29 for the rabbits. The model’s solution mirrors this structure closely. The only notable nit is a minor misattribution in the airplane part where the model references a “(iii) arc finiteness axiom” that belongs to rabbits; the paper instead proves continuity for the airplane via wings and the quotient argument (Proposition 2.11 → Propositions 2.13–2.15). Overall, the arguments align and are correct in substance. Key cites: Theorem B summarizes both identifications (as Theorems 1.29 and 2.16) ; the dendrite quotient for the airplane is Proposition 2.5 ; the embedding/surjectivity/continuity of Π are Propositions 2.13–2.15 and Theorem 2.16 ; the rabbit framework (legal colorings, tree-of-circles, and the isomorphism) is in Lemma 1.21, Proposition 1.26, and Theorem 1.29 .