Global hyperbolicity of renormalization for $C^r$ unimodal mappings

Edson de Faria¹ ; Welington de Melo² ; Alberto Pinto³

¹ Instituto de Matemática e Estatística, Universidade de São Paulo, 05508-090 Sao Paulo SP, Brazil
² I.M.P.A., Rio de Janeiro, Brazil 22460-320
³ Departamentos de Matemática, Universidade do Porto, Porto, Portugal

Annals of mathematics, Tome 164 (2006) no. 3, pp. 731-824.

Résumé

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 $r \ge 2+\alpha$ 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 holonomy is $C^{1+\beta}$ for some $\beta>0$. We also prove that the global stable sets are $C^1$ immersed (codimension one) submanifolds as well, provided $r \ge 3+\alpha$ 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.

Détail

DOI : 10.4007/annals.2006.164.731