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Transactions of the American Mathematical Society

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Models for spaces of dendritic polynomials


Authors: Alexander Blokh, Lex Oversteegen, Ross Ptacek and Vladlen Timorin
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 37F20; Secondary 37F10, 37F50
DOI: https://doi.org/10.1090/tran/7482
Published electronically: January 24, 2019
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Abstract: Complex 1-variable polynomials with connected Julia sets and only repelling periodic points are called dendritic. By results of Kiwi, any dendritic polynomial is semiconjugate to a topological polynomial whose topological Julia set is a dendrite. We construct a continuous map of the space of all cubic dendritic polynomials onto a laminational model that is a quotient space of a subset of the closed bidisk. This construction generalizes the ``pinched disk'' model of the Mandelbrot set due to Douady and Thurston. It can be viewed as a step towards constructing a model of the cubic connectedness locus.


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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@uab.edu

Ross Ptacek
Affiliation: Faculty of Mathematics, National Research University Higher School of Economics, 6 Usacheva str., Moscow, Russia, 119048
Email: rptacek@ufl.edu

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

DOI: https://doi.org/10.1090/tran/7482
Keywords: Complex dynamics, laminations, Mandelbrot set, Julia set
Received by editor(s): July 31, 2017
Received by editor(s) in revised form: November 17, 2017, and October 5, 2018
Published electronically: January 24, 2019
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–1807558
The fourth named author was supported by the Russian Academic Excellence Project ‘5-100’.
Article copyright: © Copyright 2019 American Mathematical Society