Preprint A477/2006
Chaotic Period Doubling

C. P. Tresser | V.V.M.S Chandramouli,, | M. Martens, | W. de Melo,

**Keywords: **
renormalization | hyperbolicity

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, eventually
by means that generalize to other renormalization operators. It was
then proved that also in the space of $C^{2+\alpha}$ unimodal maps,
for $\alpha$ close to one, the period doubling renormalization fixed
point is hyperbolic. Smoothness influences crucially the small
scale geometry for various types of topological dynamics, as has
been known for some time \emph{e.g.,} in Hamiltonian dynamics and
for circle maps. 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
$C^2$. What is important is that the lack of hyperbolicity is among
maps at the boundary of chaos, which is corresponds to the small
scale geometry not being map-independent. Things get much
worse with just a small loss of smoothness from $C^2$ as then, even the uniqueness is lost and other asymptotic behavior become possible. Indeed we show
that the period doubling renormalization operator acting on the space of $C^{1+Lip}$ unimodal maps has infinite entropy.