Global hyperbolicity of renormalization for C-r unimodal mappings

Abstract

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).