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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 2024 MCQ for Transactions of the American Mathematical Society is 1.48 .

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On the hyperbolicity of the period-doubling fixed point
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by Daniel Smania PDF
Trans. Amer. Math. Soc. 358 (2006), 1827-1846 Request permission

Abstract:

We give a new proof of the hyperbolicity of the fixed point for the period-doubling renormalization operator using the local dynamics near a semi-attractive fixed point (in a Banach space) and the theory of holomorphic motions. We also give a new proof of the exponential contraction of the Feigenbaum renormalization operator in the hybrid class of the period-doubling fixed point: our proof uses the non-existence of invariant line fields in the period-doubling tower (C. McMullen), the topological convergence (D. Sullivan), and a new infinitesimal argument.
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Additional Information
  • Daniel Smania
  • Affiliation: Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 1A1
  • Address at time of publication: Departamento de Matemática, ICMC-USP-Campus de São Carlos, Caixa Postal 668, São Carlos-SP, CEP 13560-970, Brazil
  • Email: smania@icmc.usp.br
  • Received by editor(s): March 19, 2003
  • Received by editor(s) in revised form: July 16, 2004
  • Published electronically: October 31, 2005
  • Additional Notes: This work was partially supported by CNPq-Brazil grant 200764/01-2, University of Toronto and USP-São Carlos.
  • © Copyright 2005 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 358 (2006), 1827-1846
  • MSC (2000): Primary 37F25, 37E20; Secondary 37F45
  • DOI: https://doi.org/10.1090/S0002-9947-05-03803-1
  • MathSciNet review: 2186998