## Institute for Mathematical Sciences

## Preprint ims01-01

** E. de Faria, W de Melo and A. Pinto**
* 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 $\mathbb{U}^r$ of $C^r$ unimodal maps with quadratic
critical point. We show that in $\mathbb{U}^r$ 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 $\mathbb{U}^r$, 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.

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