Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)

 

Puzzle geometry and rigidity: The Fibonacci cycle is hyperbolic


Author: Daniel Smania
Journal: J. Amer. Math. Soc. 20 (2007), 629-673
MSC (2000): Primary 37F30, 37C15, 30C62, 30C65, 37F25, 37F45, 37E20
Published electronically: January 17, 2007
MathSciNet review: 2291915
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We describe a new and robust method to prove rigidity results in complex dynamics. The new ingredient is the geometry of the critical puzzle pieces: under control of geometry and ``complex bounds'', two generalized polynomial-like maps which admit a topological conjugacy, quasiconformal outside the filled-in Julia set, are indeed quasiconformally conjugate. The proof uses a new abstract removability-type result for quasiconformal maps, following ideas of Heinonen and Koskela and of Kallunki and Koskela, optimized for applications in complex dynamics. We prove, as the first application of this new method, that, for even criticalities distinct from two, the period two cycle of the Fibonacci renormalization operator is hyperbolic with $ 1$-dimensional unstable manifold.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (2000): 37F30, 37C15, 30C62, 30C65, 37F25, 37F45, 37E20

Retrieve articles in all journals with MSC (2000): 37F30, 37C15, 30C62, 30C65, 37F25, 37F45, 37E20


Additional Information

Daniel Smania
Affiliation: Institute for Mathematical Sciences, State University of New York at Stony Brook, Stony Brook, New York 11794-3660
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/S0894-0347-07-00550-4
PII: S 0894-0347(07)00550-4
Keywords: Fibonacci combinatorics, generalized renormalization, puzzle, rigidity, quasiconformal conjugacy, removability, hyperbolicity
Received by editor(s): April 19, 2005
Published electronically: January 17, 2007
Additional Notes: This work was supported by grants CNPq-200764/01-2, CNPq-472316/2003-6 and FAPESP-03/03107-9. Visits to Warwick University and ICTP-Trieste were supported by CNPq-Brazil grant 460110/00-4, Warwick University and ICTP-Trieste.
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.