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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles 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 1.48.

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When do two rational functions have locally biholomorphic Julia sets?
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by Romain Dujardin, Charles Favre and Thomas Gauthier HTML | PDF
Trans. Amer. Math. Soc. 376 (2023), 1601-1624 Request permission

Abstract:

In this article we address the following question, whose interest was recently renewed by problems arising in arithmetic dynamics: under which conditions does there exist a local biholomorphism between the Julia sets of two given one-dimensional rational maps? In particular we find criteria ensuring that such a local isomorphism is induced by an algebraic correspondence. This extends and unifies classical results due to Baker, Beardon, Eremenko, Levin, Przytycki and others. The proof involves entire curves and positive currents.
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Additional Information
  • Romain Dujardin
  • Affiliation: Sorbonne Université, Laboratoire de probabilités, statistique et modélisation, 4 place Jussieu, 75005 Paris, France
  • MR Author ID: 687799
  • ORCID: 0000-0003-3798-0316
  • Email: romain.dujardin@sorbonne-universite.fr
  • Charles Favre
  • Affiliation: CMLS, Ecole Polytechnique, 91128 Palaiseau Cedex, France
  • MR Author ID: 641179
  • Email: charles.favre@polytechnique.edu
  • Thomas Gauthier
  • Affiliation: Laboratoire de Mathématiques d’Orsay, Bâtiment 307, Université Paris-Saclay, 91405 Orsay Cedex, France
  • MR Author ID: 1019319
  • Email: thomas.gauthier1@universite-paris-saclay.fr
  • Received by editor(s): January 17, 2022
  • Received by editor(s) in revised form: July 12, 2022
  • Published electronically: October 3, 2022
  • Additional Notes: The third author was partially supported by the ANR grant Fatou ANR-17-CE40-0002-01.
  • © Copyright 2022 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 376 (2023), 1601-1624
  • MSC (2010): Primary 37F10
  • DOI: https://doi.org/10.1090/tran/8775
  • MathSciNet review: 4549686