## Institute for Mathematical Sciences

## Preprint ims00-10

** A. Avila, M. Martens and W. de Melo**
* On the Dynamics of the Renormalization Operator.*

Abstract: An important part of the bifurcation diagram of unimodal maps corresponds to infinite renormalizable maps. The dynamics of
the renormalization operator describes this part of the
bifurcation pattern precisely. Here we analyze the dynamics of
the renormalization operator acting on the space of $C^k$
infinitely renormalizable maps of bounded type. We prove that
two maps of the same type are exponentially asymptotic. We
suppose $k \geq 3$ and quadratic critical point.

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