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Chaotic period doubling

Published online by Cambridge University Press:  01 April 2009

V. V. M. S. CHANDRAMOULI ,
M. MARTENS ,
W. DE MELO  and
C. P. TRESSER
V. V. M. S. CHANDRAMOULI
Affiliation:
Department of Mathematics, RUG, PO Box 407, 9700 AK Groningen, The Netherlands Department of Mathematics, SUNY at Stony Brook, Stony Brook, NY 11794-3651, USA (email: marco@math.sunysb.edu)
M. MARTENS
Affiliation:
Department of Mathematics, SUNY at Stony Brook, Stony Brook, NY 11794-3651, USA (email: marco@math.sunysb.edu)
W. DE MELO
Affiliation:
IMPA, R. Dona Castorina 110, Rio de Janeiro, Brasil
C. P. TRESSER
Affiliation:
IBM, T. J. Watson Research Center, PO Box 218, Rte 134, Yorktown Heights, NY 10598, USA
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Abstract

The period doubling renormalization operator was introduced by Feigenbaum and by Coullet and Tresser in the 1970s to study the asymptotic small-scale geometry of the attractor of one-dimensional systems that are at the transition from simple to chaotic dynamics. This geometry turns out not to depend on the choice of the map under rather mild smoothness conditions. The existence of a unique renormalization fixed point that 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 C2+α unimodal maps, for α>0, 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 result states that in the space of C2 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 C2+∣⋅∣, the failure of hyperbolicity is tamer than in C2. Things get much worse with just a bit less smoothness than C2, as then even the uniqueness is lost and other asymptotic behavior becomes possible. We show that the period doubling renormalization operator acting on the space of C1+Lip unimodal maps has infinite topological entropy.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 29 , Issue 2 , April 2009 , pp. 381 - 418
Copyright
Copyright © Cambridge University Press 2008

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