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Integrability of Hamiltonian systems on Cantor sets. (English) Zbl 0542.58015
This is an unusually well-written and well-organized paper. In an eight page ’overview’ the main result is explained in detail and put into context of the existing theory. The proof, which is inherently long and rather complicated is presented in the remaining 35 pages in a very lucid fashion by breaking the result, and the proof, into various parts, including two major subtheorems and some lemmas. The concluding section contains some further results which are related to the main one. As far as the contents is concerned, the opening sections of the overview more than adequately speak for themselves: ”Perturbations of integrable Hamiltonian systems of n degrees of freedom are in general no longer integrable - they may even become ergodic on each energy surface. However, if the integrable system is nondegenerate, the perturbed system still exhibits to a large extent quasiperiodic motions on invariant tori, as is well-known from the theory of Kolmogorov, Arnol’d and Moser. But the family of these tori is parametrized over a Cantor set and hence nowhere dense. Our aim is to show that, nonetheless, on this Cantor set these tori form a differentiable family in the sense of Whitney, which allows us to speak of an integrable system on a Cantor set. Moreover, this foliation exhibits the phenomenon of anisotropic differentiability, that is, it admits more derivatives tangential to these tori than transversal to them. Simple corollaries will be the existence of n independent functions in involution, which, on a Cantor set, form integrals of the motion, on an estimate of the measure of the complement of the invariant tori in terms of the perturbations. This provides another aspect of ’integrability’ for perturbed systems.
Traditionally the problem of integrating a Hamiltonian system was formulated as the problem of finding the complete solution for the Hamilton-Jacobi equation. We shall construct a smooth, anisotropically differentiable function, which, on a Cantor set, solves this equation for the perturbed system. This can be viewed as a third aspect of ’integrability’. We obtain our results for differentiable perturbations of a real analytic system which are of class \(C^ r\) with \(r>3n-1\) and small in a weighted \(C^ r\)-norm. Moreover, assuming additional smoothness for the Hamiltonian we get additional smoothness for the foliation and the solution of the Hamilton-Jacobi equation without any further smallness assumption, including the case of an infinitely often differentiable or even analytic Hamiltonian.”
It is remarkable how well the author has succeeded in making even the very technical parts eminently readable - while not denying the hard labour required if one wants to go into every detail.
Reviewer: H.S.P.Grässer

MSC:
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
70H05 Hamilton’s equations
58C35 Integration on manifolds; measures on manifolds
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[1] Arnol’d, Trans. A.M.S., Ser. 2 46 pp 213– (1965)
[2] Arnol’d, Uspehi. Math. Nauk. 18 pp 13– (1963)
[3] Russian Math. Surveys 18 pp 9– (1963)
[4] and , Smooth prime integrals for quasi integrable Hamiltonian systems, Typoscript, Institute Matematico, Universita di Roma, 1981.
[5] Hörmander, Ark. Mat. 3 pp 555– (1958)
[6] Katok, Math. USSR Izv. 7 pp 535– (1973)
[7] The general theory of dynamical systems and classical mechanics, Appendix D in R. Abraham, Foundations of Mechanics, Benjamin, 1967.
[8] Lazutkin, Math. USSR Izv. 7 pp 185– (1973)
[9] Montel, Bull. Soc. Math. France 46 pp 151– (1918)
[10] On invariant curves of area preserving mappings of an annulus, Nachr. Akad. Wiss. Gott., Math. Phys. Kl., 1962, pp. 1–20. · Zbl 0107.29301
[11] Moser, Ann. Sc. Norm. Sup., Pisa 20 pp 265– (1966)
[12] On the continuation of almost periodic solutions for ordinary differential equations, Proc. of Int’l. Conf. on Func. Anal. and Rel. Topics, Tokyo, 1969, pp. 60–67.
[13] Stable and Random Motions in Dynamical Systems, Princeton University Press, 1973.
[14] Approximation of Functions of Several Variables and Imbedding Theorems, Springer, 1975. · doi:10.1007/978-3-642-65711-5
[15] Les Méthodes Nouvelles de la Mécanique Celeste II, Gauthier-Villars, 1892.
[16] Pöschel, Bonn. Math. Schr. 120 (1980)
[17] Kleine Nenner I: Über invariante Kurven differenzierbarer Abbildungen eines Kreisringes, Nachr. Akad. Wiss. Gött., Math. Phys. Kl., 1970, pp. 67–105. · Zbl 0201.11202
[18] On optimal estimates for the solutions of linear partial differential equations of first order with constant coefficients on the torus, in Dynamical Systems, Theory and Applications, Lecture Notes in Physics 38, 1975, pp. 598–624. · doi:10.1007/3-540-07171-7_19
[19] and , Lectures on Celestial Mechanics, Springer, 1971. · doi:10.1007/978-3-642-87284-6
[20] Singular Integrals and Differentiability Properties of Functions, Princeton, 1970. · Zbl 0207.13501
[21] Svanidze, Tr. Mat. Inst. Steklova 147 pp 124– (1980)
[22] Whitney, Trans. A.M.S. 36 pp 63– (1934)
[23] Zehnder, Coram. Pure Appl. Math. 28 pp 91– (1975)
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