## Institute for Mathematical Sciences

## Preprint ims07-02

** V.V.M.S. Chandramouli, M. Martens, W. De Melo, C.P. Tresser**
*Chaotic Period Doubling*

Abstract: The period doubling renormalization operator was introduced by M. Feigenbaum and by P. Coullet and C.Tresser in the nineteen-seventieth to study the asymptotic small
scale geometry of the attractor of one-dimensional systems which are
at the transition from simple to chaotic dynamics. This geometry
turns out to not depend on the choice of the map under rather mild
smoothness conditions. The existence of a unique renormalization
fixed point which is also hyperbolic among generic smooth enough
maps plays a crucial role in the corresponding renormalization
theory. The uniqueness and hyperbolicity of the renormalization
fixed point were first shown in the holomorphic context, by means that
generalize to other renormalization operators. It was
then proved that in the space of $C^{2+\alpha}$ unimodal maps,
for $\alpha$ close to one, the period doubling renormalization fixed
point is hyperbolic as well. In this paper we study what happens when one
approaches from below the minimal smoothness thresholds for the
uniqueness and for the hyperbolicity of the period doubling
renormalization generic fixed point. Indeed, our main results
states that in the space of $C^2$ unimodal maps the analytic fixed
point is not hyperbolic and that the same remains true when adding
enough smoothness to get a priori bounds. In this smoother class,
called $C^{2+|\cdot|}$ the failure of hyperbolicity is tamer than in
$C^2$. Things get much
worse with just a bit less of smoothness than $C^2$ as then even the
uniqueness is lost and other asymptotic behavior become possible. We show
that the period doubling renormalization operator acting on the space of
$C^{1+Lip}$ unimodal maps has infinite topological entropy.

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