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Transactions of the American Mathematical Society

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Building blocks for Quadratic Julia sets


Authors: Joachim Grispolakis, John C. Mayer and Lex G. Oversteegen
Journal: Trans. Amer. Math. Soc. 351 (1999), 1171-1201
MSC (1991): Primary 30C35, 54F20
DOI: https://doi.org/10.1090/S0002-9947-99-02346-6
MathSciNet review: 1615975
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Abstract: We obtain results on the structure of the Julia set of a quadratic polynomial $P$ with an irrationally indifferent fixed point $z_0$ in the iterative dynamics of $P$. In the Cremer point case, under the assumption that the Julia set is a decomposable continuum, we obtain a building block structure theorem for the corresponding Julia set $J=J(P)$: there exists a nowhere dense subcontinuum $B\subset J$ such that $P(B)=B$, $B$ is the union of the impressions of a minimally invariant Cantor set $A$ of external rays, $B$ contains the critical point, and $B$ contains both the Cremer point $z_0$ and its preimage. In the Siegel disk case, under the assumption that no impression of an external ray contains the boundary of the Siegel disk, we obtain a similar result. In this case $B$ contains the boundary of the Siegel disk, properly if the critical point is not in the boundary, and $B$ contains no periodic points. In both cases, the Julia set $J$ is the closure of a skeleton $S$ which is the increasing union of countably many copies of the building block $B$ joined along preimages of copies of a critical continuum $C$ containing the critical point. In addition, we prove that if $P$ is any polynomial of degree $d\ge 2$ with a Siegel disk which contains no critical point on its boundary, then the Julia set $J(P)$ is not locally connected. We also observe that all quadratic polynomials which have an irrationally indifferent fixed point and a locally connected Julia set have homeomorphic Julia sets.


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Additional Information

Joachim Grispolakis
Affiliation: Technical University of Crete, Chania, Greece
Email: mgrysp@euclid.aml.tuc.gr

John C. Mayer
Email: mayer@math.uab.edu

Lex G. Oversteegen
Affiliation: University of Alabama at Birmingham, Birmingham, Alabama 35294-1170
Email: overstee@math.uab.edu

DOI: https://doi.org/10.1090/S0002-9947-99-02346-6
Keywords: Julia set, complex analytic dynamics, prime ends, decomposable continuum, quadratic polynomial, Siegel disk, Cremer point, irrationally indifferent fixed point, rotation number
Received by editor(s): August 30, 1995
Additional Notes: Portions of this paper were presented at the Spring Topology Conference in Auburn, Alabama, March 1994, and in the special session on Geometry of Dynamical Systems at the AMS meeting in Orlando, Florida, March 1995.
Article copyright: © Copyright 1999 American Mathematical Society

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