The Julia sets of basic uniCremer polynomials of arbitrary degree

Blokh, Alexander; Oversteegen, Lex

doi:10.1090/S1088-4173-09-00195-7

The Julia sets of basic uniCremer polynomials of arbitrary degree
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by Alexander Blokh and Lex Oversteegen

Conform. Geom. Dyn. 13 (2009), 139-159

DOI: https://doi.org/10.1090/S1088-4173-09-00195-7

Published electronically: June 17, 2009

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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.

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