Global hyperbolicity of renormalization for C-r unimodal mappings

• Type: Article in International Scientific Journal
• Date: 2006

Authors

Edson De Faria (Author) Other; Welington De Melo (Author) Other; Alberto Pinto (Author) FCUP

Journal

Title: Annals of Mathematics

Vol. 164

Pages: 731-824

ISSN: 0003-486X

Publisher: Princeton University Press

Indexing

ISI Web of Knowledge - 37 Citations

Scopus - 40 Citations

Scientific classification

FOS: Natural sciences > Mathematics

Other information

Authenticus ID: 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).

Language: English

Type (Professor's evaluation): Scientific

No. of pages: 94