Orbit portraits of unicritical antiholomorphic polynomials

Abstract

Abstract. Orbit portraits were introduced by Goldberg and Milnor as a combinatorial tool to describe the patterns of all periodic dynamical rays landing on a periodic cycle of a quadratic polynomial. This encodes information about the dynamics and the parameter spaces of these maps. We carry out a similar analysis for unicritical antiholomorphic polynomials, and give an explicit description of the orbit portraits that can occur for such maps in terms of their characteristic angles, which turns out to be rather restricted when compared with the holomorphic case. Finally, we prove a realization theorem for these combinatorial objects. The results obtained in this paper serve as a combinatorial foundation for a detailed understanding of the combinatorics and topology of the parameter spaces of unicritical antiholomorphic polynomials and their connectedness loci, known as the multicorns.

On multicorns and unicorns II: bifurcations in spaces of antiholomorphic polynomials

“…Arguing as in Lemma 4.10, we see that four distinct dynamical rays Rc α 1 , Rc α 2 , Rc α ′ 1 , and Rc α ′ 2 (each of period 2k) land at the characteristic parabolic point of fc. This contradicts [Mu,Lemma 2.10]. Hence, C 1 ∩ C 2 = ∅.…”

“…Let z c be the characteristic parabolic point of f c . Since the dynamical ray R c ϑ lands at the k-periodic point z c , by [Mu,Lemma 2.5], the period of ϑ (under multiplication by −d) is either k or 2k.…”

“…Let the period of H be k. The common period of α 1 and α 2 under multiplication by −d is 2k. Any accumulation point of these two parameter rays is either a parabolic parameter of odd period k or a parabolic parameter of even period r with r|2k such that the corresponding dynamical ray of period 2k lands at the characteristic parabolic point in the dynamical plane of that parameter (by Lemma 4.5 and [Mu,Lemma 2.5]). Define the set F to be the union of the closure of the finitely many root arcs and cusp points of period k, and the finitely many parabolic parameters of even parabolic period and of ray period 2k (this is precisely the set of all parameters c for which f c has a parabolic cycle such that a dynamical ray R c t of period 2k may land at the parabolic cycle of f c ).…”

“…Pick c ′ ∈ U 1 \ M * d such that t(c ′ ) ∈ (α 1 , α 2 ). By [Mu,Lemma 3.4], the dynamical rays R c ′ α 1 and R c ′ α 2 land at a common repelling periodic point of f c ′ . Since U 1 ⊂ C \ t∈A P R t ∪ F , it follows from Lemma 2.4 that R c α 1 and R c α 2 land at a common repelling periodic point of f c for all c in U 1 .…”

“…If the two parameter rays R α 1 and R α 2 do not land at the same point or do not accumulate on the closure of the same root arc (recall that every hyperbolic component of odd period has at most one root arc, and any two root arcs have disjoint closures), then U 1 would contain parameters c outside M * d with t(c) / ∈ (α 1 , α 2 ). But it follows from the remark at the end of the proof of [Mu,Theorem 3.1] that for such a parameter c, the dynamical rays R c α 1 and R c α 2 can not land at a common point, which is a contradiction to the stability of the ray landing pattern throughout U i . Hence, the parameter rays R α 1 and R α 2 must either land at the same parabolic parameter of even (parabolic) period and of ray period 2k, or accumulate on the closure of the same root arc/cusp of M * d .…”