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Complementary components to the cubic principal hyperbolic domain


Authors: Alexander Blokh, Lex Oversteegen, Ross Ptacek and Vladlen Timorin
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
Published electronically: August 7, 2018
MathSciNet review: 3856134
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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\vert _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
Email: ablokh@math.uab.edu

Lex Oversteegen
Affiliation: Department of Mathematics, University of Alabama at Birmingham, Birmingham, Alabama 35294
Email: overstee@math.uab.edu

Ross Ptacek
Affiliation: Department of Mathematics, University of Alabama at Birmingham, Birmingham, Alabama 35294
Email: rptacek@uab.edu

Vladlen Timorin
Affiliation: National Research University Higher School of Economics, 6 Usacheva ul., 119048 Moscow, Russia
Email: vtimorin@hse.ru

DOI: https://doi.org/10.1090/proc/14168
Keywords: Complex dynamics, Julia set, polynomial-like maps, laminations
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
Article copyright: © Copyright 2018 American Mathematical Society