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Renormalization of vector fields and Diophantine invariant tori

Published online by Cambridge University Press:  01 October 2008

HANS KOCH  and
SAŠA KOCIĆ
HANS KOCH
Affiliation:
Department of Mathematics, University of Texas at Austin, 1 University Station C1200, Austin, TX 78712, USA (email: koch@math.utexas.edu, kocic@math.utexas.edu)
SAŠA KOCIĆ
Affiliation:
Department of Mathematics, University of Texas at Austin, 1 University Station C1200, Austin, TX 78712, USA (email: koch@math.utexas.edu, kocic@math.utexas.edu)
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Abstract

We extend the renormalization group techniques that were developed originally for Hamiltonian flows to more general vector fields on 𝕋d×ℝℓ. Each Diophantine vector ω∈ℝd determines an analytic manifold 𝒲 of infinitely renormalizable vector fields, and each vector field on 𝒲 is shown to have an elliptic invariant d-torus with frequencies ω1,ω2,…,ωd. Analogous manifolds for particular classes of vector fields (Hamiltonian, divergence-free, symmetric, reversible) are obtained simply by restricting 𝒲 to the corresponding subspace. We also discuss non-degeneracy conditions, and the resulting reduction in the number of parameters needed in parametrized families to guarantee the existence of invariant tori.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 28 , Issue 5 , October 2008 , pp. 1559 - 1585
Copyright
Copyright © 2008 Cambridge University Press

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References

[1]Abad, J. J. and Koch, H.. Renormalization and periodic orbits for Hamiltonian flows. Commun. Math. Phys. 212 (2000), 371–394.CrossRefGoogle Scholar
[2]Broer, H. W.. KAM theory: the legacy of Kolmogorov’s 1954 paper. Bull. Amer. Math. Soc. 41 (2004), 507–521.CrossRefGoogle Scholar
[3]Broer, H. W., Huitema, G. B. and Sevryuk, M. B.. Quasi-Periodicity in Families of Dynamical Systems: Order Amidst Chaos (Lecture Notes in Mathematics, 1645). Springer, Berlin, 1996.Google Scholar
[4]Broer, H. W., Huitema, G. B. and Takens, F.. Unfoldings of quasi-periodic tori. Mem. Amer. Math. Soc. 83 (1990), 1–82.Google Scholar
[5]Cassels, J. W. S.. An Introduction to Diophantine Approximation. Cambridge University Press, Cambridge, 1957.Google Scholar
[6]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
[7]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
[8]de la Llave, R.. A Tutorial on KAM Theory (Proceedings of Symposia in Pure Mathematics, 69). Eds. A. Katok et al. American Mathematical Society, Providence, RI, 2001, pp. 175–292.Google Scholar
[9]Eliasson, L. H.. Absolutely convergent series expansions for quasi periodic motions. Report 2–88, Department of Mathematics, University of Stockholm, 1988. Math. Phys. Electron J. 2(4) (1996), 33pp.Google Scholar
[10]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
[11]Gaidashev, D. G.. Renormalization of isoenergetically degenerate Hamiltonian flows and associated bifurcations of invariant tori. Discrete Contin. Dynam. Systems A 13 (2005), 63–102.CrossRefGoogle Scholar
[12]Gallavotti, G. and Gentile, G.. Hyperbolic low-dimensional invariant tori and summations of divergent series. Commun. Math. Phys. 227 (2002), 421–460.CrossRefGoogle Scholar
[13]Gallavotti, G., Gentile, G. and Mastropietro, V.. Field theory and KAM tori. Math. Phys. Electron J. 1(5) (1995), 13pp..Google Scholar
[14]Gentile, G., Bartuccelli, M. V. and Deane, J. H. B.. Summation of divergent series and Borel summability for strongly dissipative differential equations with periodic or quasiperiodic forcing terms. J. Math. Phys. 46 (2005), 062704, 21pp.CrossRefGoogle Scholar
[15]Gentile, G. and Mastropietro, V.. Methods for the analysis of the Lindstedt series for KAM tori and renormalizability in classical mechanics. A review with some applications. Rev. Math. Phys. 8 (1996), 393–444.CrossRefGoogle Scholar
[16]Khanin, K., Lopes Dias, J. and Marklof, J.. Multidimensional continued fractions, dynamic renormalization and KAM theory. Commun. Math. Phys. 270 (2007), 197–231.CrossRefGoogle Scholar
[17]Khanin, K. and Sinai, Ya.. The renormalization group method and Kolmogorov–Arnold–Moser theory. Nonlinear Phenomena in Plasma Physics and Hydrodynamics. Ed. R. Z. Sagdeev. Mir, Moscow, 1986, pp. 93–118.Google Scholar
[18]Kleinbock, D. Y. and Margulis, G. A.. Flows on homogeneous spaces and Diophantine approximation on manifolds. Ann. of Math. (2) 148 (1998), 339–360.CrossRefGoogle Scholar
[19]Koch, H.. A renormalization group for Hamiltonians, with applications to KAM tori. Ergod. Th. & Dynam. Sys. 19 (1999), 1–47.CrossRefGoogle Scholar
[20]Koch, H.. A renormalization group fixed point associated with the breakup of golden invariant tori. Discrete Contin. Dynam. Systems 11 (2004), 881–909.CrossRefGoogle Scholar
[21]Koch, H.. Existence of critical invariant tori. Ergod. Th. & Dynam. Sys. to appear.Google Scholar
[22]Koch, H. and Lopes Dias, J.. Renormalization of Diophantine skew flows, with applications to the reducibility problem. Discrete Contin. Dynam. Systems A 21 (2008), 477–500.CrossRefGoogle Scholar
[23]Kocić, S.. Renormalization of Hamiltonians for Diophantine frequency vectors and KAM tori. Nonlinearity 18 (2005), 2513–2544.CrossRefGoogle Scholar
[24]Lagarias, J. C.. Geodesic multidimensional continued fractions. Proc. London Math. Soc. 69 (1994), 464–488.CrossRefGoogle Scholar
[25]Lopes Dias, J.. Renormalisation scheme for vector fields on T 2 with a Diophantine frequency. Nonlinearity 15 (2002), 665–679.CrossRefGoogle Scholar
[26]Lopes Dias, J.. Brjuno condition and renormalization for Poincaré flows. Discrete Contin. Dynam. Systems 15 (2006), 641–656.CrossRefGoogle Scholar
[27]MacKay, R. S.. Three Topics in Hamiltonian Dynamics (Dynamical Systems and Chaos, 2). Eds. Y. Aizawa, S. Saito and K. Shiraiwa. World Scientific, London, 1995.Google Scholar
[28]Moser, J.. Convergent series expansions for quasi-periodic motions. Math. Ann. 169 (1967), 136–176.CrossRefGoogle Scholar
[29]Sevryuk, M. B.. Reversible Systems (Lecture Notes in Mathematics, 1211). Springer, Berlin, 1986.CrossRefGoogle Scholar