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Conformal Geometry and Dynamics

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The Julia sets of basic uniCremer polynomials of arbitrary degree

Authors: Alexander Blokh and Lex Oversteegen
Journal: Conform. Geom. Dyn. 13 (2009), 139-159
MSC (2000): Primary 37F10; Secondary 37F50, 37B45, 37C25, 54F15
Published electronically: June 17, 2009
MathSciNet review: 2511916
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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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Additional Information

Alexander Blokh
Affiliation: Department of Mathematics, University of Alabama at Birmingham, Birmingham, Alabama 35294-1170

Lex Oversteegen
Affiliation: Department of Mathematics, University of Alabama at Birmingham, Birmingham, Alabama 35294-1170

Keywords: Complex dynamics, Julia set, Cremer fixed point
Received by editor(s): May 8, 2008
Published electronically: June 17, 2009
Additional Notes: The first author was partially supported by NSF grant DMS-0456748
The second author was partially supported by NSF grant DMS-0405774
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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