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Existence of critical invariant tori

Published online by Cambridge University Press:  23 October 2008

HANS KOCH
HANS KOCH*
Affiliation:
Department of Mathematics, The University of Texas at Austin, Austin, TX 78712, USA
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Abstract

We consider analytic Hamiltonian systems with two degrees of freedom, and prove that every Hamiltonian on the strong local stable manifold of the renormalization group fixed point obtained in Koch [A renormalization group fixed point associated with the breakup of golden invariant tori. Discrete Contin. Dyn. Syst. A 11 (2004), 881–909] has a non-differentiable golden invariant torus (conjugacy to a linear flow).

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 28 , Issue 6 , December 2008 , pp. 1879 - 1894
Copyright
Copyright © 2008 Cambridge University Press

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References

[1] Escande, D. F. and Doveil, F.. Renormalisation method for computing the threshold of the large scale stochastic instability in two degree of freedom Hamiltonian systems. J. Stat. Phys. 26 (1981), 257–284.CrossRefGoogle Scholar
[2] Kadanoff, L. P.. Scaling for a critical Kolmogorov–Arnold–Moser trajectory. Phys. Rev. Lett. 47 (1981), 1641–1643.CrossRefGoogle Scholar
[3] Shenker, S. J. and Kadanoff, L. P.. Critical behavior of a KAM surface. I. Empirical results. J. Stat. Phys. 27 (1982), 631–656.CrossRefGoogle Scholar
[4] MacKay, R. S.. Renormalisation in area preserving maps. Thesis, Princeton, 1982 (unpublished). World Scientific, London, 1993.CrossRefGoogle Scholar
[5] Mehr, A. and Escande, D. F.. Destruction of KAM tori in Hamiltonian systems: link with the destabilization of nearby cycles and calculation of residues. Phys. D 13 (1984), 302–338.CrossRefGoogle Scholar
[6] Khanin, K. and Sinai, Ya. G.. The renormalization group method and KAM theory. Nonlinear Phenomena in Plasma Physics and Hydrodynamics. Ed. R. Z. Sagdeev. Mir, Moscow, 1986, pp. 93–118.Google Scholar
[7] Wilbrink, J.. New fixed point of the renormalization operator for invariant circles. Phys. Lett. A 131 (1988), 251–255.CrossRefGoogle Scholar
[8] Khanin, K. and Sinai, Ya. G.. Renormalization group methods in the theory of dynamical systems. Internat. J. Modern Phys. B 2 (1988), 147–165.Google Scholar
[9] Wilbrink, J.. New fixed point of the renormalisation operator associated with the recurrence of invariant circles in generic Hamiltonian maps. Nonlinearity 3 (1990), 567–584.CrossRefGoogle Scholar
[10] Kosygin, D.. Multidimensional KAM theory from the renormalization group viewpoint. Dynamical Systems and Statistical Mechanics (AMS, Advances in Soviet Mathematics, 3). Ed. Ya. G. Sinai. 1991, pp. 99–129.CrossRefGoogle Scholar
[11] MacKay, R. S., Meiss, J. D. and Stark, J.. An approximate renormalization for the break-up of invariant tori with three frequencies. Phys. Lett. A 190 (1994), 417–424.CrossRefGoogle Scholar
[12] Stirnemann, A.. Towards an existence proof of Mackay’s fixed point. Comm. Math. Phys. 188 (1997), 723–735.CrossRefGoogle Scholar
[13] Chandre, C., Govin, M. and Jauslin, H. R.. KAM-renormalization group analysis of stability in Hamiltonian flows. Phys. Rev. Lett. 79 (1997), 3881–3884.Google Scholar
[14] Del-Castillo-Negrete, D., Greene, J. M. and Morrison, P. J.. Renormalization and transition to chaos in area preserving non-twist maps. Phys. D 100 (1997), 311–329.CrossRefGoogle Scholar
[15] Chandre, C., Govin, M., Jauslin, H. R. and Koch, H.. Universality for the breakup of invariant tori in Hamiltonian flows. Phys. Rev. E 57 (1998), 6612–6617.Google Scholar
[16] Chandre, C., Jauslin, H. R., Benfatto, G. and Celletti, A.. An approximate renormalization-group transformation for Hamiltonian systems with three degrees of freedom. Phys. Rev. E 60 (1999), 5412–5421.Google ScholarPubMed
[17] Chandre, C. and MacKay, R. S.. Approximate renormalization with codimension-one fixed point for the break-up of some three-frequency tori. Phys. Lett. A 275 (2000), 394–400.CrossRefGoogle Scholar
[18] Lopes Dias, J.. Renormalisation of flows on the multidimensional torus close to a KT frequency vector. Nonlinearity 15 (2002), 647–664.CrossRefGoogle Scholar
[19] Lopes Dias, J.. Renormalisation scheme for vector fields on T 2 with a Diophantine frequency. Nonlinearity 15 (2002), 665–679.CrossRefGoogle Scholar
[20] Chandre, C. and Jauslin, H. R.. Renormalization-group analysis for the transition to chaos in Hamiltonian systems. Phys. Rep. 365 (2002), 1–64.CrossRefGoogle Scholar
[21] Apte, A., Wurm, A. and Morrison, P. J.. Renormalization and destruction of 1/γ 2 tori in the standard nontwist map. Chaos 13 (2003), 421–433.CrossRefGoogle ScholarPubMed
[22] Abad, J. J., Koch, H. and Wittwer, P.. A renormalization group for Hamiltonians: numerical results. Nonlinearity 11 (1998), 1185–1194.CrossRefGoogle Scholar
[23] Koch, H.. A renormalization group for Hamiltonians, with applications to KAM tori. Ergod. Th. & Dynam. Sys. 19 (1999), 1–47.Google Scholar
[24] Abad, J. J. and Koch, H.. Renormalization and periodic orbits for Hamiltonian flows. Comm. Math. Phys. 212 (2000), 371–394.Google Scholar
[25] Koch, H.. On the renormalization of Hamiltonian flows, and critical invariant tori. Discrete Contin. Dyn. Syst. A 8 (2002), 633–646.CrossRefGoogle Scholar
[26] Koch, H.. A renormalization group fixed point associated with the breakup of golden invariant tori. Discrete Contin. Dyn. Syst. A 11 (2004), 881–909.CrossRefGoogle Scholar
[27] Gaidashev, D. G.. Renormalization of isoenergetically degenerate Hamiltonian flows and associated bifurcations of invariant tori. Discrete Contin. Dyn. Syst. A 13 (2005), 63–102.CrossRefGoogle Scholar
[28] Kocić, S.. Renormalization of Hamiltonians for Diophantine frequency vectors and KAM tori. Nonlinearity 18 (2005), 2513–2544.CrossRefGoogle Scholar
[29] de la Llave, R. and Olvera, A.. The obstruction criterion for non existence of invariant circles and renormalization. Nonlinearity 19(8) (2006), 1907–1937.CrossRefGoogle Scholar
[30] Ada files and data can be found online at http://journals.cambridge.org/ETS; see also ftp://ftp.ma.utexas.edu/pub/papers/koch/nonsmooth/.Google Scholar
[31] Yampolsky, M.. Hyperbolicity of renormalization of critical circle maps. Publ. Math. Inst. Hautes Etudes Sci. 96 (2002), 1–41.Google Scholar
[32] Yampolsky, M.. Siegel disks and renormalization fixed points. Field Inst. Commun. 53 (2008).Google Scholar
[33] Hartman, P.. On local homeomorphisms of Euclidean spaces. Bol. Soc. Mat. Mexicana 5 (1960), 220–241.Google Scholar
[34] Hirsch, M. W., Pugh, C. C. and Shub, M.. Invariant Manifolds (Lecture Notes in Mathematics, 583). Springer, Berlin, 1977.CrossRefGoogle Scholar
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