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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On the hyperbolicity of the period-doubling fixed point


Author: Daniel Smania
Journal: Trans. Amer. Math. Soc. 358 (2006), 1827-1846
MSC (2000): Primary 37F25, 37E20; Secondary 37F45
Published electronically: October 31, 2005
MathSciNet review: 2186998
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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

DOI: http://dx.doi.org/10.1090/S0002-9947-05-03803-1
PII: S 0002-9947(05)03803-1
Keywords: Renormalization, parabolic domain, petals, holomorphic motion, Feigenbaum, period-doubling, universality, semi-attractive, hyperbolicity
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.
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.