Abstract
The disadvantage of ‘traditional’ multidimensional continued fraction algorithms is that it is not known whether they provide simultaneous rational approximations for generic vectors. Following ideas of Dani, Lagarias and Kleinbock-Margulis we describe a simple algorithm based on the dynamics of flows on the homogeneous space \(SL(d, \mathbb{Z}) \backslash SL(d, \mathbb{R})\) (the space of lattices of covolume one) that indeed yields best possible approximations to any irrational vector. The algorithm is ideally suited for a number of dynamical applications that involve small divisor problems. As an example, we explicitly construct a renormalization scheme for the linearization of vector fields on tori of arbitrary dimension.
Similar content being viewed by others
References
Abad J.J., Koch H. (2000) Renormalization and periodic orbits for Hamiltonian flows. Commun. Math. Phys. 212: 371–394
Ávila A., Krikorian R.: Reducibility or non-uniform hyperbolicity for quasiperiodic Schrödinger cocycles. Ann. of Math., to appear
Bricmont J., Gawȩdzki K., Kupiainen A. (1999) KAM theorem and quantum field theory. Commun. Math. Phys. 201, 699–727
Cassels J.W.S. (1957) An Introduction to Diophantine Approximation. Cambridge, Cambridge University Press
Dani S.G. (1985) Divergent trajectories of flows on homogeneous spaces and Diophantine approximation. J. Reine Angew. Math. 359, 55–89
Feigenbaum M.J. (1978) Quantitative universality for a class of non-linear transformations. J. Stat. Phys. 19, 25–52
Gaidashev D.G. (2005) Renormalization of isoenergetically degenerate Hamiltonian flows and associated bifurcations of invariant tori. Discrete Contin. Dyn. Syst. 13, 63–102
Gallavotti G. (1994) Twistless KAM tori. Commun. Math. Phys. 164, 145–156
Gentile G., Mastropietro V. (1996) 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, 393–444
Grabiner D.J., Lagarias J.C. (2001) Cutting sequences for geodesic flow on the modular surface and continued fractions. Monatsh. Math. 133, 295–339
Hardcastle D.M. (2002) The three-dimensional Gauss algorithm is strongly convergent almost everywhere. Experiment. Math. 11, 131–141
Hardcastle D.M., Khanin K. (2002) The d-dimensional Gauss transformation: strong convergence and Lyapunov exponents. Experiment. Math. 11, 119–129
Herman M.R. (1989) Inégalités “a priori” pour des tores lagrangiens invariants par des difféomorphismes symplectiques. Inst. Hautes Études Sci. Publ. Math. 70, 47–101
Hille, E., Phillips, R. S.: Functional analysis and semi-groups, Volume 31. AMS Colloquium Publications, Rev. ed. of 1957, Providence, RI:Amer. Math. soc., 1974
Khanin, K., Lopes Dias, J., Marklof, J.: Renormalization of multidimensional Hamiltonian flows. Nonlinearity (2006) (to appear) Preprint, 2005, available at http://pascal.iseg.utl.pt/~jldias/KLM.pdf
Khanin K., Sinai Ya (1986) The renormalization group method and Kolmogorov-Arnold-Moser theory. In: Sagdeev R.Z. (eds) Nonlinear phenomena in plasma physics and hydrodynamics. Moscow, Mir, pp. 93–118
Kleinbock D.Y., Margulis G.A. (1998) Flows on homogeneous spaces and Diophantine approximation on manifolds. Ann. of Math 148: 339–360
Koch H. (1999) A renormalization group for Hamiltonians, with applications to KAM tori. Erg. Theor. Dyn. Syst. 19, 475–521
Koch H. (2002) On the renormalization of Hamiltonian flows, and critical invariant tori. Discrete Contin. Dyn. Syst. 8(3): 633–646
Koch H. (2004) A renormalization group fixed point associated with the breakup of golden invariant tori. Discrete Contin. Dyn. Syst. 11, 881–909
Lagarias J.C. (1994) Geodesic multidimensional continued fractions. Proc. London Math. Soc. 69, 464–488
Lopes Dias J. (2002) Renormalization of flows on the multidimensional torus close to a KT frequency vector. Nonlinearity 15, 647–664
Lopes Dias J. (2002) Renormalization scheme for vector fields on \(\mathbb{T}^{2}\) with a diophantine frequency. Nonlinearity 15, 665–679
Lopes Dias J. (2006) Brjuno condition and renormalization for Poincaré flows. Discrete Contin. Dyn. Syst. 15, 641–656
MacKay R.S. (1993) Renormalisation in area-preserving maps. River Edge, NJ: World Scientific Publishing Co. Inc.,
MacKay R.S. (1995) Three topics in Hamiltonian dynamics. In: Aizawa Y., Saito S., Shiraiwa K. (eds) Dynamical Systems and Chaos, Volume 2. Singapore, World Scientific
Moore C.C. (1966) Ergodicity of flows on homogeneous spaces. Am. J. Math. 88, 154–178
Moser J. (1973) Stable and random motions in dynamical systems. Annals Math. Studies. Princeton, NJ: Princeton Univ. Press
Raghunathan M.S. (1972) Discrete subgroups of Lie groups. Berlin-Heidelberg-New York, Springer-Verlag
Schweiger F. (2000) Multidimensional continued fractions. Oxford Science Publications. Oxford, Oxford University Press
Yoccoz, J.-C.: Petits diviseurs en dimension 1 (Small divisors in dimension one). Astérisque 231, 1995
Yoccoz J.-C. (2002) Analytic linearization of circle diffeomorphisms. In: Dynamical systems and small divisors, Eds. J. Marmi, C. Yoccoz, Lecture Notes in Mathematics, 1784, Berlin-Heidelberg-New York: Springer-Verlag
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by G. Gallavotti
An erratum to this article is available at http://dx.doi.org/10.1007/BFb0120366.
Rights and permissions
About this article
Cite this article
Khanin, K., Dias, J.L. & Marklof, J. Multidimensional Continued Fractions, Dynamical Renormalization and KAM Theory. Commun. Math. Phys. 270, 197–231 (2007). https://doi.org/10.1007/s00220-006-0125-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-006-0125-y