×

Simple proofs and extensions of a result of L. D. Pustylnikov on the nonautonomous Siegel theorem. (English) Zbl 1385.37022

L. D. Pustyl’nikov [Math. USSR, Sb. 23, 382–404 (1975; Zbl 0324.34051)] extended to the nonautonomous case the Siegel theorem of linearization of analytic mappings. Five proofs of a generalization of this result are presented in this paper. More precisely, if a sequence \(f_n\) of analytic mappings of \({\mathbb{C}}^d\) has a common fixed point \(f_n(0)=0\), and the maps \(f_n\) converge to a linear mapping \(A_\infty=\text{diag}(e^{2\pi i\omega_1}, \ldots , e^{2\pi i\omega_d})\), \(\omega = (\omega_1, \ldots, \omega_d)\in{\mathbb{R}}^d\), so fast that \[ \sum_n \|f_n-A_\infty\|_{L^\infty(B)}<\infty\,, \] then \(f_n\) is nonautonomously conjugate to the linearization. That is, there exists a sequence \(h_n\) of analytic mappings fixing the origin satisfying \(h_{n+1}\circ f_n=A_\infty\,h_n\). The key point is that the functions \(h_n\) are defined in a large domain and they are bounded. It is shown that \(\sum_n \|h_n- \text{Id}\|_{L^\infty(B)}<\infty\).
Some results are also provided in the case in which \(f_n\) converges to a nonlinearizable map \(f_\infty\) or to a nonelliptic linear mapping. When the mappings \(f_n\) preserve a geometric structure (for instance, symplectic, volume, contact) the same happens for the chosen \(h_n\).
One proof is based on the Cook method of scattering theory; another proof uses just the elementary implicit function theorem; the third proof is only based on compactness arguments; the fourth proof is based on a very simple Nash-Moser theorem which takes advantage of a very subtle cancellation; and finally, the fifth proof is based on the deformation method.

MSC:

37B55 Topological dynamics of nonautonomous systems
37C60 Nonautonomous smooth dynamical systems
37F50 Small divisors, rotation domains and linearization in holomorphic dynamics
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
47J07 Abstract inverse mapping and implicit function theorems involving nonlinear operators

Citations:

Zbl 0324.34051
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Abraham, R. and Marsden, J.E., Foundations of Mechanics, Reading, Mass.: Benjamin/Cummings, 1978.
[2] Arnol’d, V. I., Geometrical Methods in the Theory of Ordinary Differential Equations, 2nd ed., Grundlehren Math. Wiss., vol. 250, New York: Springer, 1988. · Zbl 0648.34002
[3] Astakhov, S.A.; Lee, E. A.; Farrelly, D., Formation of Kuiper-belt binaries through multiple chaotic scattering encounters with low-mass intruders, Mon. Not. R. Astron. Soc., 360, 401-415, (2005)
[4] Banyaga, A.; Llave, R.; Wayne, C.E., Cohomology equations near hyperbolic points and geometric versions of sternberg linearization theorem, J. Geom. Anal., 6, 613-649, (1996) · Zbl 0918.53016
[5] Bartsch, Th.; Revuelta, F.; Benito, R.M.; Borondo, F., Reaction rate calculation with time-dependent invariant manifolds, J. Chem. Phys., 136, 224510, (2012)
[6] Blazevski, D.; Llave, R., Time-dependent scattering theory for ODEs and applications to reaction dynamics, J. Phys. A, 44, 195101, (2011) · Zbl 1223.34119
[7] Brjuno, A.D., Analytic form of differential equations: 1, Trans. Moscow Math. Soc., 25, 131-288, (1971)
[8] Brjuno, A.D., Analytic form of differential equations: 2, Trans. Moscow Math. Soc., 26, 199-239, (1972)
[9] Bruno, A.D., Local Methods in Nonlinear Differential Equations, Berlin: Springer, 1989.
[10] Buslaev, V.; Pushnitski, A., The scattering matrix and associated formulas in Hamiltonian mechanics, Comm. Math. Phys., 293, 563-588, (2010) · Zbl 1218.37072
[11] Calleja, R.; Llave, R., A numerically accessible criterion for the breakdown of quasi-periodic solutions and its rigorous justification, Nonlinearity, 23, 2029-2058, (2010) · Zbl 1209.37008
[12] Canadell, M.; Llave, R., KAM tori andwhiskered invariant tori for non-autonomous systems, Phys. D, 310, 104-113, (2015) · Zbl 1364.37048
[13] Llave, R.; Marco, J.M.; Moriyón, R., Canonical perturbation theory of Anosov systems and regularity results for the livˇsic cohomology equation, Ann. of Math. (2), 123, 537-611, (1986) · Zbl 0603.58016
[14] de la Llave, R., A Tutorial on KAM Theory, in Smooth Ergodic Theory and Its Applications (Seattle, Wash., 1999), A. Katok, R. de la Llave, Ya. Pesin, H. Weiss (Eds.), Proc. Sympos. Pure Math., vol. 69, Providence,R.I.: AMS, 2001, pp. 175-292.
[15] de la Llave, R., Uniform Boundedness of Analytic Iterates Implies Linearizability: A Simple Proof and Extensions: Preprint (2017).
[16] de la Llave, R., Introduction to KAM Theory, Izhevsk: Institute of Computer Science, 2003 (Russian).
[17] DeLatte, D., Nonstationary normal forms and cocycle invariants, Random Comput. Dynam., 1, 229-259, (1992) · Zbl 0778.58058
[18] DeLatte, D. and Gramchev, T., Biholomorphic Maps with Linear Parts Having Jordan Blocks: Linearization and Resonance Type Phenomena, Math. Phys. Electron. J., 2002, vol. 8, Paper 2, 27 pp. · Zbl 1038.37038
[19] Dereziński, J. and Gérard, Ch., Scattering Theory of Classical and Quantum N-Particle Systems, Texts Monogr. Phys., Berlin: Springer, 1997. · Zbl 0899.47007
[20] Dieudonné, J., Foundations of Modern Analysis, Pure Appl. Math., vol. 10, New York: Acad. Press, 1960. · Zbl 0100.04201
[21] Guysinsky, M., The theory of non-stationary normal forms, Ergodic Theory Dynam. Systems, 22, 845-862, (2002) · Zbl 1010.37026
[22] Hale, J.K., Ordinary Differential Equations, 2nd ed., Huntington,N.Y.: Krieger, 1980. · Zbl 0433.34003
[23] Herman, M.-R., Recent results and some open questions on siegel’s linearization theorem of germs of complex analytic diffeomorphisms of \(C\)\^{}{n} near a fixed point, 138-184, (1987)
[24] Krüger, T.; Pustyl’nikov, L.D.; Troubetzkoy, S., The nonautonomous function-theoretic center problem, Bol. Soc. Brasil. Mat. (N. S.), 30, 1-30, (1999) · Zbl 0930.37003
[25] Mather, J., Anosov diffeomorphisms, Bull. Amer. Math. Soc., 73, 792-795, (1967)
[26] Meyer, K. R., The implicit function theorem and analytic differential equations, 191-208, (1975)
[27] Moser, J., On a theorem of Anosov, J. Differential Equations, 5, 411-440, (1969) · Zbl 0169.42303
[28] Moser, J., On the volume elements on a manifold, Trans. Amer. Math. Soc., 120, 286-294, (1965) · Zbl 0141.19407
[29] Moser, J., A rapidly convergent iteration method and non-linear partial differential equations: 1, Ann. Scuola Norm. Sup. Pisa (3), 20, 265-315, (1966) · Zbl 0144.18202
[30] Nelson, E., Topics in Dynamics: 1. Flows. Mathematical Notes, Princeton,N.J.: Princeton Univ. Press, 1969. · Zbl 0197.10702
[31] Pustyl’nikov, L.D., A generalization of two theorems of C. L. Siegel to the nonautonomous case, Uspehi Mat. Nauk, 26, 245-246, (1971) · Zbl 0232.30007
[32] Pustyl’nikov, L.D., Stable and oscillating motions in nonautonomous dynamical systems: A generalization of C.L. siegel’s theorem to the nonautonomous case, Math. USSR-Sb., 23, 382-404, (1974) · Zbl 0324.34051
[33] Reed, M. and Simon, B., Methods of Modern Mathematical Physics: 3. Scattering Theory, New York: Acad. Press, 1979. · Zbl 0405.47007
[34] Saks, S. and Zygmund, A., Analytic Functions, 2nd ed., enl., Monogr. Matem., vol. 28, Warsaw: PWN, 1965. · Zbl 0136.37301
[35] Siegel, C. L., Iteration of analytic functions, Ann. of Math. (2), 43, 607-612, (1942) · Zbl 0061.14904
[36] Simon, B., Wave operators for classical particle scattering, Comm. Math. Phys., 23, 37-48, (1971) · Zbl 0238.70012
[37] Smale, S., Differentiable dynamical systems, Bull. Amer. Math. Soc., 73, 747-817, (1967) · Zbl 0202.55202
[38] Sternberg, Sh., Infinite Lie groups and the formal aspects of dynamical systems, J. Math. Mech., 10, 451-474, (1961) · Zbl 0131.26802
[39] Thirring, W., Classical Mathematical Physics: Dynamical Systems and Field Theories, 3rd ed., New York: Springer, 1997.
[40] Levine, H., Singularities of differentiable mappings, 1-21, (1971) · Zbl 0216.45803
[41] Wheeler, J.A., On the mathematical description of light nuclei by the method of resonating group structure, Phys. Rev., 52, 1107-1122, (1937) · JFM 63.1405.01
[42] Yomdin, Y., Nonautonomous linearization, 718-726, (1988) · Zbl 0666.58014
[43] Zehnder, E., A simple proof of a generalization of a theorem by C. L. Siegel, 855-866, (1977)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.