Available in electronic format
Available in print format		 Transacrions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(e) ISSN 0002-9947(p)
Previous issue | Table of contents | Next issue
Articles in press | Recently posted articles | Most recent issue | All issues
Previous Article | Next Article
     

On the hyperbolicity of the period-doubling fixed point

Author(s): Daniel Smania
Journal: Trans. Amer. Math. Soc. 358 (2006), 1827-1846.
MSC (2000): Primary 37F25, 37E20; Secondary 37F45
Posted: October 31, 2005
Retrieve article in: PDF DVI PostScript

Abstract | References | Similar articles | Additional information

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.

References:

[AB]
L. Ahlfors and L. Bers.
Riemann's mapping theorem for variable metrics.
Ann. of Math., 72:385-404, 1960. MR 0115006 (22:5813)

[ALdM]
A. Avila, M. Lyubich and W. de Melo.
Regular or stochastic dynamics in real analytic families of unimodal maps.
Invent. Math. 154:451-550, 2003. MR 2018784

[DH]
A. Douady and J. Hubbard.
On the dynamics of polynomial-like mappings.
Ann. Sci. École Norm. Sup (4), 18:287-343, 1985. MR 0816367 (87f:58083)

[H]
M. Hakim.
Attracting domains for semi-attractive transformations of $ C\sp p$.
Publ. Mat. 38, 479-499, 1994. MR 1316642 (96a:32050)

[Lan]
O. Lanford.
A computer-assisted proof of the Feigenbaum conjectures.
Bull. Amer. Math. Soc. 6:427-434, 1982. MR 0648529 (83g:58051)

[Lyu97]
M. Lyubich.
Dynamics of quadratic polynomials, I-II.
Acta Math., 178:185-297, 1997. MR 1459261 (98e:58145)

[Lyu99]
M. Lyubich.
Feigenbaum-Coullet-Tresser universality and Milnor's hairness conjecture.
Ann. of Math., 149:319-420, 1999. MR 1689333 (2000d:37051)

[Lyu02]
M. Lyubich.
Almost every real quadratic map is either regular or stochastic.
Ann. of Math., 156:1-78, 2002. MR 1935840 (2003i:37032)

[McM94]
C. McMullen.
Complex dynamics and renormalization.
Annals of Mathematics Studies, 135, Princeton University Press, Princenton, 1994.MR 1312365 (96b:58097)

[McM96]
C. McMullen.
Renormalization and $ 3$-manifolds which fiber over the circle.
Annals of Mathematics Studies, 142, Princeton University Press, Princenton, 1996.MR 1401347 (97f:57022)

[M]
R. Mañé.
Lyapounov exponents and stable manifolds for compact transformations.
in Geometric dynamics (Rio de Janeiro, 1981), Lectures Notes in Math. 1007, 522-577. MR 0730286 (85j:58126)

[MSS]
R. Mañé; P. Sad and D. Sullivan.
On the dynamics of rational maps.
Ann. Sci. Ecole Norm. Sup., 16:193-217, 1983. MR 0732343 (85j:58089)

[Sm01]
D. Smania.
Renormalization theory for multimodal maps.
Eletronic Preprint, IMPA, 2001. Avaliable at www.icmc.usp.br/$ \sim$smania/.

[Sm02a]
D. Smania.
Puzzle geometry and rigidity: The Fibonacci cycle is hyperbolic.
Eletronic Preprint, 2002. Avaliable at www.icmc.usp.br/$ \sim$smania/.

[Sm03]
D. Smania.
On Feigenbaum eigenvectors.
In preparation, 2003.

[Su]
D. Sullivan.
Bounds, quadratic differentials and renormalization conjectures.
in American Mathematical Society Centennial publications: Vol. II (Providence, RI, 1988, 417-466, Amer. Math. Soc, Providence, 1992. MR 1184622 (93k:58194)

Similar Articles:

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 37F25, 37E20, 37F45

Retrieve articles in all Journals with MSC (2000): 37F25, 37E20, 37F45

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: 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
Posted: 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 of article: Copyright 2005, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2008, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google