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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Complementary components to the cubic principal hyperbolic domain
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by Alexander Blokh, Lex Oversteegen, Ross Ptacek and Vladlen Timorin PDF
Proc. Amer. Math. Soc. 146 (2018), 4649-4660 Request permission

Abstract:

We study the closure of the cubic Principal Hyperbolic Domain and its intersection $\mathcal {P}_\lambda$ with the slice $\mathcal {F}_\lambda$ of the space of all cubic polynomials with fixed point $0$ defined by the multiplier $\lambda$ at $0$. We show that any bounded domain $\mathcal {W}$ of $\mathcal {F}_\lambda \setminus \mathcal {P}_\lambda$ consists of $J$-stable polynomials $f$ with connected Julia sets $J(f)$ and is either of Siegel capture type (then $f\in \mathcal {W}$ has an invariant Siegel domain $U$ around $0$ and another Fatou domain $V$ such that $f|_V$ is two-to-one and $f^k(V)=U$ for some $k>0$) or of queer type (then a specially chosen critical point of $f\in \mathcal {W}$ belongs to $J(f)$, the set $J(f)$ has positive Lebesgue measure, and it carries an invariant line field).
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Additional Information
  • Alexander Blokh
  • Affiliation: Department of Mathematics, University of Alabama at Birmingham, Birmingham, Alabama 35294
  • MR Author ID: 196866
  • Email: ablokh@math.uab.edu
  • Lex Oversteegen
  • Affiliation: Department of Mathematics, University of Alabama at Birmingham, Birmingham, Alabama 35294
  • MR Author ID: 134850
  • Email: overstee@math.uab.edu
  • Ross Ptacek
  • Affiliation: Department of Mathematics, University of Alabama at Birmingham, Birmingham, Alabama 35294
  • MR Author ID: 1076403
  • Email: rptacek@uab.edu
  • Vladlen Timorin
  • Affiliation: National Research University Higher School of Economics, 6 Usacheva ul., 119048 Moscow, Russia
  • MR Author ID: 645829
  • Email: vtimorin@hse.ru
  • Received by editor(s): November 11, 2014
  • Received by editor(s) in revised form: December 1, 2015, and January 20, 2016
  • Published electronically: August 7, 2018
  • Additional Notes: The first and third named authors were partially supported by NSF grant DMS–1201450.
    The second named author was partially supported by NSF grant DMS-0906316.
    The fourth named author was partially supported by the Russian Academic Excellence Project 5-100.
  • Communicated by: Nimish Shah
  • © Copyright 2018 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 146 (2018), 4649-4660
  • MSC (2010): Primary 37F45; Secondary 37F10, 37F20, 37F50
  • DOI: https://doi.org/10.1090/proc/14168
  • MathSciNet review: 3856134