Chaotic Period Doubling

TL;DR

This paper investigates how the hyperbolicity and uniqueness of the period doubling renormalization fixed point change as the smoothness of unimodal maps decreases, revealing loss of hyperbolicity and increased complexity in less smooth classes.

Contribution

It analyzes the effects of reduced smoothness on the hyperbolicity and uniqueness of the renormalization fixed point, extending the understanding beyond the classical smooth settings.

Findings

In $C^2$ maps, the fixed point is not hyperbolic.

In $C^{2+| ext{·}|}$ class, hyperbolicity failure is milder.

In $C^{1+Lip}$ maps, the operator has infinite topological entropy.

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.