Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-24T18:43:41.200Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Avila, A., Martens, M. and de Melo, W.. On the Dynamics of the Renormalization Operator. Global Analysis of Dynamical Systems, Festschift dedicated to Floris Takens 60th birthday. Institute of Physics, Bristol, 2001.CrossRefGoogle Scholar
[2]Arnol’d, V. I.. Small denominators, I: mappings of the circumference onto itself. Amer. Math. Soc. Transl. 46 (1965), 213–284.Google Scholar
[3]Birkhoff, G., Martens, M. and Tresser, C.. On the scaling structure for period doubling. Astérisque 286 (2003), 167–186.Google Scholar
[4]Coullet, P. and Tresser, C.. Itération d’endomorphismes et groupe de renormalisation. J. Phys. Colloque C5 (1978), C5-25–C5-28.Google Scholar
[5]Davie, A. M.. Period doubling for C 2+ϵ mappings. Comm. Math. Phys. 176 (1999), 262–272.Google Scholar
[6]Epstein, H.. Fixed points of composition operators II. Nonlinearity 2 (1989), 305–310.CrossRefGoogle Scholar
[7]Eckmann, J. P. and Wittwer, P.. A complete proof of the Feigenbaum conjectures. J. Stat. Phys. 46 (1987), 455–475.CrossRefGoogle Scholar
[8]Feigenbaum, M. J.. Quantitative universality for a class of non-linear transformations. J. Stat. Phys. 19 (1978), 25–52.CrossRefGoogle Scholar
[9]Feigenbaum, M. J.. The universal metric properties of nonlinear transformations. J. Stat. Phys. 21 (1979), 669–706.CrossRefGoogle Scholar
[10]de Faria, E., de Melo, W. and Pinto, A.. Global hyperbolicity of renormalization for C r unimodal mappings. Ann. of Math. (2) 164 (2006).CrossRefGoogle Scholar
[11]Herman, M. R.. Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations. Publ. Math. Inst. Hautes Études Sci. 49 (1979), 5–233.CrossRefGoogle Scholar
[12]Katznelson, Y. and Ornstein, D.. The differentiability of the conjugation of certain diffeomorphisms of the circle. Ergod. Th. & Dynam. Sys. 9 (1989), 643–680.CrossRefGoogle Scholar
[13]Khanin, K. M. and Sinai, Ya. G.. A new proof of M. Herman’s theorem. Comm. Math. Phys. 112 (1987), 89–101.CrossRefGoogle Scholar
[14]Lanford III, O. E.. A computer assisted proof of the Feigenbaum conjecture. Bull. Amer. Math. Soc. (N.S.) 6 (1984), 427–434.CrossRefGoogle Scholar
[15]Lyubich, M.. Feigenbaum–Coullet–Tresser universality and Milnor’s hairiness conjecture. Ann. of Math. (2) 149 (1999), 319–420.CrossRefGoogle Scholar
[16]Ma, S. K.. Modern Theory of Critical Phenomena. Benjamin, Reading, MA, 1976.Google Scholar
[17]Martens, M.. The periodic points of renormalization. Ann. of Math. (2) 147 (1998), 543–584.CrossRefGoogle Scholar
[18]Martens, M.. Distortion results and invariant Cantor sets of unimodal maps. Ergod. Th. & Dynam. Sys. 14 (1994), 331–349.CrossRefGoogle Scholar
[19]Martens, M., de Melo, W., Van Strien, S. and Sullivan, D.. Bounded geometry and measure of the attracting Cantor set of quadratic-like interval maps. Preprint, June 1988.Google Scholar
[20]de Melo, W. and van Strien, S.. One-Dimensional Dynamics. Springer, Berlin, 1993.CrossRefGoogle Scholar
[21]McMullen, C.. Complex Dynamics and Renormalization (Annals of Mathematics Studies, 135). Princeton University Press, Princeton, NJ, 1994.Google Scholar
[22]Sullivan, D.. Bounds, Quadratic Differentials, and Renormalization Conjectures (Mathematics into the Twenty-First Century, 2). American Mathematical Society, Providence, RI, 1992, American Mathematical Society Centennial Publication.Google Scholar
[23]Tresser, C.. Fine structure of universal Cantor sets. Instabilities and Nonequilibrium Structures III. Eds. E. Tirapegui and W. Zeller. Kluwer, Dordrecht, 1991.Google Scholar
[24]Tresser, C. and Coullet, P.. Itérations d’endomorphismes et groupe de renormalisation. C. R. Acad. Sci. Paris 287A (1978), 577–580.Google Scholar
[25]Yoccoz, J.-C.. Conjugaison différentiable des difféomorphismes du cercle dont le nombre de rotation vérifie une condition diophantienne. Ann. Sci. École Norm. Sup. (4) 17 (1984), 333–359.CrossRefGoogle Scholar