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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
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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.

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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

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.

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