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.