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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