## 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