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References
[Arn] Arnold, V.I.:Proof of a theorem by A.N. Kolmogorov on the invariance of quasi-periodic motions under small perturbations of the Hamiltonian. Usp. Math. Nauk18, 13–40 (1963); Russ. Math. Surv.18, 9–36 (1963)
[Brj] Brjuno, A.D.:Analytic form of differential equations. Trans. Mosc. Math. Soc.25, 131–288 (1971)
[CC] Celletti, A., Chierchia, L.:Construction of analytic KAM surfaces and effective stability bounds. Commun. Math. Phys.118, 119–161 (1988)
[DRV] Dodson, M.M., Rynne, B.P., Vickers, J.A.G.:Metric diophantine approximation and Hausdorff dimensions on manifolds. Math. Proc. Camb. Philos. Soc. (to appear)
[Eli] Eliasson, L.H.:Perturbations of stable invariant tori. Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser.15, 115–147 (1988)
[ESW] Fröhlich, J., Spencer, T., Wayne, C.E.:Localization in disordered, nonlinear dynamical systems. J. Stat. Phys.42, 247–274 (1986)
[Gra] Graff, S.M.:On the continuation of hyperbolic invariant tori for hamiltonian systems. Jour. Differ. Equations.15, 1–69 (1974)
[Kol] Kolmogrov, A.N.:On the conservation of conditionally periodic motions for a small change in Hamilton's function. Dokl. Akad. Nauk. SSSR98, 525–530 (1954)
[Koz] Kozlov, S.M.:Reducibility of quasi-periodic differential operators and averaging. Trans. Mosc. Math. Soc.46, 101–126 (1984)
[Kuk] Kuksin, S.B.:Hamiltonian perturbations of infinite-dimensional linear systems with an imaginary spectrum. Funkts. Anal. Prilozh.21, 22–37 (1987); Funct. Anal. Appl.21, 192–205 (1988)
[Mel-1] Melnikov, V.K.:On some cases of conservation of conditionally periodic motions under a small change of the Hamiltonian function. Sov. Math., Dokl.6, 1592–1596 (1965)
[Mel-2] Melnikov, V.K.:A family of conditionally periodic solutions of a Hamiltonian system. Sov. Math. Dokl.9, 882–886 (1968)
[Mos-1] Moser, J.:On invariant curves of area preserving mappings of an annulus. Nachr. Akad. Wiss. Gött., Math.-Phys. Kl. 1–20 (1962)
[Mos-2] Moser, J.:Convergent series expansions for quasi-periodic motions. Math. Ann.169, 136–176 (1967)
[Mos-3] Moser, J.:A stability theorem for minimal foliations on a torus. Ergodic Theory Dyn. Syst.8, 251–281 (1988)
[Nik] Nikolenko, N.V.:The method of Poincaré normal forms in problems of integrability of equations of evolution type. Russ. Math. Surv.41, 63–114 (1986)
[Pös-1] Pöschel, J.:Integrability of Hamiltonian systems on Cantor sets. Commun. Pure Appl. Math.35, 653–695 (1982)
[Pös-2] Pöschel, J.:On invariant manifolds of complex analytic mappings near fixed points. Expo. Math.4, 97–109 (1986)
[Pös-3] Pöschel, J.:A general infinite dimensional KAM-theorem. In:Proceedings of the IX International Congress on Mathematical Physics, Swansea (1988)
[Pös-4] Pöschel, J.:Small divisors with spatial structure. Bonn University, 1989 (preprint)
[PT] Pöschel, J., Trubowitz, E.:Inverse Spectral Theory. Boston: Academic Press (1987)
[Rüs-1] Rüssmann, H.:On the one-dimensional Schrödinger equation with a quasi-periodic potential. Annals of the New York Acad. Sci.357, 90–107 (1980)
[Rüs-2] Rüssmann, H.: Talk held at the ETH Zürich February 1986
[Sie-1] Siegel, C.L.:Iteration of analytic functions. Ann. Math.43, 607–612 (1942)
[Sie-2] Siegel, C.L.:Über die Normalform analytischer Differentialgleichungen in der Nähe einer Gleichgewichtslösung. Nachr. Akad. Wiss. Göttingen, Math.-Phys. Kl. 21–30 (1952)
[SM] Siegel, C.L., Moser, J.:Lectures on Celestial Mechanics. Berlin-Heidelberg-New York: Springer, (1971)
[Spr] Sprindžuk, V.G.:Theory of Diophantine Approximations. New York: Wiley (1979)
[St] Stein, E.:Singular Integrals and Differentiability Properties of Functions. Princeton: Princeton University Press (1970)
[SZ] Salamon, D., Zehnder, E.:KAM-theory in configuration space. Comment. Math. Helv.64, 84–132 (1989)
[Vit] Vittot, M.:Théorie classique des perturbations et grand nombre de degres de liberté. Thèse de doctorat de l'universitè de Provence 1985
[VB] Vittot, M., Bellissard, J.:Invariant tori for an infinite lattice of coupled classical rotators. CPT-Marseille, 1985 (preprint)
[War] Ware, B.:Infinite-dimensional versions of two theorems by Carl Siegel. Bull. Am. Math. Soc.82, 613–615 (1976)
[Way] Wayne, C.E.:Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory. Preprint No 88027, Penn. State University, 1988
[Whit] Whitney, H.:Analytic extensions of differentiable functions defined in closed sets. Trans. A.M.S.36, 63–89 (1934)
[Zeh-1] Zehnder, E.:Generalized implicit function theorems with applications to some small divisor problems, I and II. Commun. Pure Appl. Math.28, 91–140 (1975); 49–111 (1976)
[Zeh-2] Zehnder, E.:Siegel's linearization theorem in infinite dimensions. Manus. math.23, 363–371 (1978)
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Partly supported by the U.S. Army Research Office through the Mathematical Sciences Institute of Cornell University and the Sonderforschungsbereich 256 at the University of Bonn
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Pöschel, J. On elliptic lower dimensional tori in hamiltonian systems. Math Z 202, 559–608 (1989). https://doi.org/10.1007/BF01221590
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DOI: https://doi.org/10.1007/BF01221590