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Transactions of the American Mathematical Society

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ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Building blocks for quadratic Julia sets
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by Joachim Grispolakis, John C. Mayer and Lex G. Oversteegen PDF
Trans. Amer. Math. Soc. 351 (1999), 1171-1201 Request permission

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
  • MR Author ID: 134850
  • Email: overstee@math.uab.edu
  • 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.
  • © Copyright 1999 American Mathematical Society
  • 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