The Julia sets of basic uniCremer polynomials of arbitrary degree

Abstract

Abstract. Let P be a polynomial of degree d with a Cremer point p and no repelling or parabolic periodic bi-accessible points. We show that there are two types of such Julia sets J P . The red dwarf J P are nowhere connected im kleinen and such that the intersection of all impressions of external angles is a continuum containing p and the orbits of all critical images. The solar J P are such that every angle with dense orbit has a degenerate impression disjoint from other impressions and J P is connected im kleinen at its landing point. We study bi-accessible points and locally connected models of J P and show that such sets J P appear through polynomial-like maps for generic polynomials with Cremer points. Since known tools break down for d > 2 (if d > 2, it is not known if there are small cycles near p, while if d = 2, this result is due to Yoccoz), we introduce wandering ray continua in J P and provide a new application of Thurston laminations.

Locally connected models for Julia sets

“…Proof. As explained in Subsection 5.3, the construction and the arguments similar to those from Theorem 4.2 [6] imply that there is a (possibly) bigger than K ′ but still wandering ray-continuum K (with the same eventual images as K ′ ) whose grand orbit Γ (i.e. the collection of pullbacks of its forward images) is well-defined.…”

“…, m. We will call K a wandering collection if distinct forward images of continua K i are all pairwise disjoint. By the arguments similar those from Theorem 4.2 [6] one can associate to K a geometric prelamination L K generated by K, and then its closure -a geo-lamination L K generated by K. For completeness we will briefly explain the main ideas of this theorem.…”

“…have more than 2 d angles (and therefore they are closed). Now it is not hard to show (see Theorem 4.2 [6]) that the family of convex hulls of so defined sets of angles H K ′ , K ′ ∈ Γ form a geometric prelamination which we denote by L K . By the definition each original ray-continuum K i has the associated to it set of angles A i , and it follows that…”

“…A polynomial P is said to be basic uniCremer if it has a Cremer fixed point and no repelling/parabolic periodic point of P is bi-accessible (a point is called bi-accessible if at least two rays land it). In this case the only monotone map of J P onto a locally connected continuum is a collapse of J P to a point [4,5,6], strongly contrasting with [13].…”

Non-degenerate quadratic laminations

“…Geometric laminations serve as a tool for studying non-locally connected Julia sets [17]. An advantage of considering them is that a geometric lamination can be constructed if only one (but an appropriately chosen one) of its leaves is known.…”