Global hyperbolicity of renormalization for C^r unimodal mappings

• Autores: Edson de Faria, Wellington de Melo, Alberto Pinto
• Localización: Annals of mathematics, ISSN 0003-486X, Vol. 164, Nº 3, 2006, págs. 731-824
• Idioma: inglés
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• Resumen

  • In this paper we extend M. Lyubich¡¯s recent results on the global hyperbolicity of renormalization of quadratic-like germs to the space of Cr 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 ¡Ý 2+¿Á with ¿Á 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 C1 codimension one, Banach submanifolds of the ambient space, and whose holonomy is C1+¿Â for some ¿Â > 0. We also prove that the global stable sets are C1 immersed (codimension one) submanifolds as well, provided r ¡Ý 3 + ¿Á with ¿Á close to one. As a corollary, we deduce that in generic, oneparameter 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.