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A KAM phenomenon for singular holomorphic vector fields. (English) Zbl 1114.37026

The classical KAM theorem deals with completely integrable Hamiltonian vector fields and their Hamiltonian perturbations. A completely integrable Hamiltonian vector field on a compact symplectic manifold induces a foliation by invariant tori of half-dimension and defines a quasiperiodic motion on each torus. Roughly, speaking, the KAM theorem says that if the nonperturbed (completely integrable) vector field is nondegenerate in some sense and the frequencies of some quasiperiodic motion satisfy a Diophantine condition, then the perturbed field (which is not completely integrable in general) has a positive measure set of invariant tori with quasiperiodic motions [A. N. Kolmogorov, “On the preservation of conditionally periodic motions under small variations of the Hamilton function”, Dokl. Akad. Nauk SSSR 98, 527–530 (1954; Zbl 0056.31502), English translation in Selected works of A. N. Kolmogorov. Volume I: Mathematics and mechanics. Mathematics and Its Applications, Soviet Series, 25. Dordrecht etc.: Kluwer Academic Publishers (1991; Zbl 0732.01045); V. I. Arnold, “Proof of a theorem of A. N. Kolmogorov on the persistence of quasi-periodic motions under small perturbations of the Hamiltonian”, Russ. Math. Surv. 18, No. 5, 9–36 (1963); translation from Usp. Mat. Nauk 18, No. 5(113), 13–40 (1963; Zbl 0129.16606); “Small denominators and the problem of stability of motion in the classical and celestial mechanics”, Russ. Math. Surv. 18, No. 6, 85–191 (1963); translation from Usp. Mat. Nauk 18, No. 6(114), 91–192 (1963; Zbl 0135.42701); J. Moser, Nachr. Akad. Wiss. Göttingen, II. Math.-Phys. Kl. 1962, 1–20 (1962; Zbl 0107.29301)].
The paper under review discovers a very interesting analogue of the KAM phenomena for appropriate local holomorphic vector fields: persistence of invariant holomorphic submanifolds under “good” perturbation. Using this result, he recovers the classical KAM theorem for real local analytic Hamiltonian vector fields (by complexification) at the end of the paper.
The statement of the main result of the paper deals with a tuple \(S_1,\dots,S_l\) of linear diagonal vector fields \(S_j\) in \(\mathbb C^n\) that satisfy a Diophantine condition and admit “sufficiently many” common first integrals (see loc. cit. below). The ring of formal first integrals is generated by monomials \(x^{R_1},\dots,x^{R_p}\). As was shown in the previous papers of the author [L. Stolovitch, C. R. Acad. Sci., Paris, Sér. I, Math. 326, No. 6, 733–736 (1998; Zbl 0917.32029), Publ. Math., Inst. Hautes Étud. Sci. 91, 133–210 (2000; Zbl 0997.32024)], any nonlinear deformation of a vector field \(S_j\) of the tuple is locally biholomorphically conjugated (near 0) to a linear combination
\[ X_0=\sum_j a_j(x^{R_1},\dots,x^{R_{p}})S_j. \]
The vector field \(X_0\) (which plays the role of a completely integrable Hamiltonian system) has first integrals \(x^{R_j}\). Hence, the fibers of the mapping \(\pi:\mathbb C^n\to \mathbb C^{p}\): \(x\mapsto(x^{R_1},\dots,x^{R_p})\) are invariant surfaces for \(X_0\) (they play the role of invariant tori).
The author fixes arbitrary \(X_0\) as above that satisfies the nondegeneracy assumption: the image of the mapping \(a=(a_1(u),\dots,a_l(u))\) is not contained in a hyperplane.
The main result of the paper under review is stated for a good perturbation \(X\) of the vector field \(X_0\), which is obtained by adding small terms of degree high enough. It says that a positive measure of the previous invariant surfaces persist: – there exist a neighborhood of zero \(U\subset\mathbb C^n\) and a positive measure set \(K\subset\mathbb C^{p}\) such that for any \(b\in K\) there exists a biholomorphism of each connected component of the fiber \(\pi^{-1}(b)\cap U\) onto an invariant holomorphic surface of the perturbed field \(X\);
– the motion of \(X\) on the invariant surface is conjugated to a linear motion on the previous fiber.
For the proof of the main result, the author makes an inductive construction of a decreasing sequence of appropriate compact sets \(K_0\supset K_1\supset K_2\dots\) in the image of \(\pi\): for any \(b\in K_k\) the restriction of \(X\) to appropriate neighborhood of the fiber \(\pi^{-1}(b)\) is “conjugated with estimates” (depending on \(k\)) to a vector field close to a “integrable” one (i.e., preserving the fibers of \(\pi\)). He shows that the intersection \(K=\cap_kK_k\) satisfies the statement of the main result. It is shown that if the previously-mentioned Diophantine condition (on the linear vector fields \(S_j\)) holds true, and if the measure of the initial compact \(K_0\) is positive, then the measure of the set \(K\) is also positive.

MSC:

37F75 Dynamical aspects of holomorphic foliations and vector fields
32M25 Complex vector fields, holomorphic foliations, \(\mathbb{C}\)-actions
32S65 Singularities of holomorphic vector fields and foliations
34M35 Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms
37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
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References:

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