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Chaotic Period Doubling | | V.V.M.S. Chandramouli
; M. Martens
; W. de Melo
; C.P. Tresser
; | | Date: |
3 Oct 2007 | | 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. | | Source: | arXiv, 0710.0667 | | Services: | Forum | Review | PDF | Favorites | |
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