The Julia sets of basic uniCremer polynomials of arbitrary degree

Abstract

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