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In this paper we extend M. Lyubich’s recent results on the global hyperbolicity of renormalization of quadratic-like germs to the space of C r unimodal maps with quadratic critical point. We show that in this space the boundedtype limit sets of the renormalization operator have an invariant hyperbolic structure provided r ≥ 2 + α with α close to one. | Annals of Mathematics Global hyperbolicity of renormalization for Cr unimodal mappings By Edson de Farian Welington de Melon and Alberto Pinto Annals of Mathematics 164 2006 731 824 Global hyperbolicity of renormalization for Cr unimodal mappings By Edson de Faria Welington de Melo and Alberto Pinto Abstract In this paper we extend M. Lyubich s recent results on the global hyper-bolicity of renormalization of quadratic-like germs to the space of Cr unimodal maps with quadratic critical point. We show that in this space the bounded-type limit sets of the renormalization operator have an invariant hyperbolic structure provided r 2 O with O close to one. As an intermediate step between Lyubich s results and ours we prove that the renormalization operator is hyperbolic in a Banach space of real analytic maps. We construct the local stable manifolds and prove that they form a continuous lamination whose leaves are C 1 codimension one Banach submanifolds of the ambient space and whose holonomy is C1 for some Ị3 0. We also prove that the global stable sets are C1 immersed codimension one submanifolds as well provided r 3 a with a close to one. As a corollary we deduce that in generic one-parameter families of Cr unimodal maps the set of parameters corresponding to infinitely renormalizable maps of bounded combinatorial type is a Cantor set with Hausdorff dimension less than one.1 Table of Contents 1. Introduction 2. Preliminaries and statements of results 2.1. Quadratic unimodal maps 2.1.1. The Banach spaces Ar 2.1.2. The Banach spaces Br 2.2. The renormalization operator 2.3. The limit sets of renormalization Financially supported by CNPq Grant 301970 2003-3. Financially supported by CNPq Grant 304912 2003-4 and Faperj Grant E-26 152.189 2002. Financially supported by Calouste Gulbenkian Foundation PRODYN-ESF POCTI and POSI by FCT and Ministerio da CTES and CMUP. There is a list of symbols used in this paper before the references for the convenience of the reader. 732 .
