  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.
