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Puzzle Geometry and Rigidity: The Fibonacci Cycle Is Hyperbolic
"... 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 polynomiallike maps which admits a topological conjugacy, quasiconformal outside ..."
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Cited by 8 (4 self)
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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 polynomiallike maps which admits a topological conjugacy, quasiconformal outside the lledin Julia set are, indeed, quasiconformally conjugated. The proof uses a new abstract removabilitytype result for quasiconformal maps, following ideas of Heinonen
On The Hyperbolicity Of The PeriodDoubling Fixed Point
 Trans. Amer. Math. Soc
, 2006
"... We give a new proof of the hyperbolicity of the fixed point for the perioddoubling renormalization operator using the local dynamics near a semiattractive fixed point (in a Banach space) and the theory of holomorphic motions. We also give a new proof of the exponential contraction of the Feigenbau ..."
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Cited by 6 (3 self)
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We give a new proof of the hyperbolicity of the fixed point for the perioddoubling renormalization operator using the local dynamics near a semiattractive 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 perioddoubling fixed point: such proof uses the non existence of invariant line fields in the perioddoubling tower (C. McMullen), the topological convergence (D. Sullivan), and a new infinitesimal argument. 1.
PHASE SPACE UNIVERSALITY FOR MULTIMODAL MAPS
"... Abstract. We study the dynamics of the renormalization operator for multimodal maps. In particular, we develop a combinatorial theory for certain kind of multimodal maps. We also prove that renormalizations of infinitely renormalizable multimodal maps with same bounded combinatorial type are expo ..."
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Cited by 1 (1 self)
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Abstract. We study the dynamics of the renormalization operator for multimodal maps. In particular, we develop a combinatorial theory for certain kind of multimodal maps. We also prove that renormalizations of infinitely renormalizable multimodal maps with same bounded combinatorial type are exponentially close. Our results imply, for instance, the existence and uniqueness of periodic points for the renormalization operator with arbitrary combinatorial type. 1.
On the Hyperbolicity of the Feigenbaum Fixed Point
"... We show the hyperbolicity of the Feigenbaum xed point using the inexibility of the Feigenbaum tower, the Man~eSadSullivan  Lemma and the existence of parabolic domains (petals) for semiattractive xed points. ..."
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We show the hyperbolicity of the Feigenbaum xed point using the inexibility of the Feigenbaum tower, the Man~eSadSullivan  Lemma and the existence of parabolic domains (petals) for semiattractive xed points.