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Transactions of the American Mathematical Society

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

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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Models for spaces of dendritic polynomials
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by Alexander Blokh, Lex Oversteegen, Ross Ptacek and Vladlen Timorin PDF
Trans. Amer. Math. Soc. 372 (2019), 4829-4849 Request permission

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
  • 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@uab.edu
  • Ross Ptacek
  • Affiliation: Faculty of Mathematics, National Research University Higher School of Economics, 6 Usacheva str., Moscow, Russia, 119048
  • MR Author ID: 1076403
  • Email: rptacek@ufl.edu
  • Vladlen Timorin
  • Affiliation: Faculty of Mathematics, National Research University Higher School of Economics, 6 Usacheva str., Moscow, Russia, 119048
  • MR Author ID: 645829
  • Email: vtimorin@hse.ru
  • 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’.
  • © Copyright 2019 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 372 (2019), 4829-4849
  • MSC (2010): Primary 37F20; Secondary 37F10, 37F50
  • DOI: https://doi.org/10.1090/tran/7482
  • MathSciNet review: 4009397