Global hyperbolicity of renormalization for C-r unimodal mappings

• Tipo: Artigo em Revista Científica Internacional
• Data: 2006

Autores

Edson De Faria (Autor) Outra; Welington De Melo (Autor) Outra; Alberto Pinto (Autor) FCUP

Revista

Título: Annals of Mathematics

Vol. 164

Páginas: 731-824

ISSN: 0003-486X

Editora: Princeton University Press

Indexação

ISI Web of Knowledge - 37 Citações

Scopus - 40 Citações

Classificação Científica

FOS: Ciências exactas e naturais > Matemática

Outras Informações

ID Authenticus: P-004-G5W

DOI: 10.4007/annals.2006.164.731

Abstract (EN): 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 bounded-type limit sets of the renormalization operator have an invariant hyperbolic structure provided r >= 2 + alpha with alpha 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 holonom is C1+beta for some beta > 0. We also prove that the global stable sets are C-1 immersed (codimension one) submanifolds as well, provided r >= 3 + alpha with alpha close to one. As a corollary, we deduce that in generic, one-parameter families of C-r 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).

Idioma: Inglês

Tipo (Avaliação Docente): Científica

Nº de páginas: 94