zbMATH — the first resource for mathematics

Simultaneous reduction to normal forms of commuting singular vector fields with linear parts having Jordan blocks. (English. French summary) Zbl 1137.37025
Summary: We study the simultaneous linearizability of \(d\)-actions (and the corresponding \(d\)-dimensional Lie algebras) defined by commuting singular vector fields in \(\mathbb C^{^{ n }}\) fixing the origin with nontrivial Jordan blocks in the linear parts. We prove the analytic convergence of the formal linearizing transformations under a certain invariant geometric condition for the spectrum of \(d\) vector fields generating a Lie algebra. If the condition fails and if we consider the situation where small denominators occur, then we show the existence of divergent solutions of an overdetermined system of linearized homological equations. In the \(\mathbb C^{^{ \infty }}\) category, the situation is completely different. We show Sternberg’s theorem for a commuting system of \(\mathbb C^{^{ \infty }}\) vector fields with a Jordan block although they do not satisfy the condition.
37F75 Dynamical aspects of holomorphic foliations and vector fields
37F50 Small divisors, rotation domains and linearization in holomorphic dynamics
37G05 Normal forms for dynamical systems
Full Text: DOI Numdam EuDML
[1] Abate, M., Diagonalization of nondiagonalizable discrete holomorphic dynamical systems, Amer. J. Math., 122, 757-781, (2000) · Zbl 0966.32018
[2] Arnold, V. I., Geometrical Methods in the Theory of Ordinary Differential Equations, (1983), Springer · Zbl 0507.34003
[3] Bruno, A. D., The analytic form of differential equations, Tr. Mosk. Mat. O-va, 25, 119-262, (1971) · Zbl 0263.34003
[4] Bruno, A. D.; Walcher, S., Symmetries and convergence of normalizing transformations, J. Math. Anal. Appl., 183, 571-576, (1994) · Zbl 0804.34040
[5] Carletti, T., Exponentially long time stability for non-linearizable analytic germs of \(({\mathbb{C}}^n,0),\) Ann Inst. Fourier (Grenoble), 54, 4, 989-1004, (2004) · Zbl 1063.37043
[6] Carletti, T.; Marmi, S., Linearization of analytic and non-analytic germs of diffeomorphisms of \(({\mathbb{C}},0),\) Bull. Soc. Math. France, 128, 69-85, (2000) · Zbl 0997.37017
[7] Chen, K. T., Diffeomorphisms:\( C^∞ -\) realizations of formal properties, Amer. J. Math., 87, 140-157, (1965) · Zbl 0151.32001
[8] Cicogna, G.; Gaeta, G., Symmetry and perturbation theory in nonlinear dynamics, 57, (1999), Springer-Verlag · Zbl 1059.37044
[9] Cicogna, G.; Walcher, S., Convergence of normal form transformations: the role of symmetries. symmetry and perturbation theory, Acta Math. Appl., 70, 95-111, (2002) · Zbl 1013.34033
[10] De La Llave, R., A tutorial on KAM theory., (1999), Univ. of Washington, Seattle · Zbl 1055.37064
[11] DeLatte, D.; Gramchev, T., Biholomorphic maps with linear parts having Jordan blocks: linearization and resonance type phenomena, Math. Physics Electronic Journal, 8, paper n. 2, 1-27, (2002) · Zbl 1038.37038
[12] Dickinson, D.; Gramchev, T.; Yoshino, M., Perturbations of vector fields on tori: resonant normal forms and Diophantine phenomena, Proc. Edinb. Math. Soc. (2), 45, 3, 731-159, (2002) · Zbl 1032.37010
[13] Dumortier, F.; Roussarie, R., Smooth linearization of germs of \(R^2\)-actions and holomorphic vector fields, Ann. Inst. Fourier (Grenoble), 30, 1, 31-64, (1980) · Zbl 0418.58015
[14] Gantmacher, F. R., The theory of matrices, 1-2, (1959), Chelsea Publishing Co., New York · Zbl 0927.15001
[15] Gramchev, T., On the linearization of holomorphic vector fields in the Siegel domain with linear parts having nontrivial Jordan blocks, World Scientific, 106-115, (2003)
[16] Herman, M., Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Publ. Math. I.H.É.S., 49, 5-233, (1979) · Zbl 0448.58019
[17] Katok, A.; Katok, S., Higher cohomology for abelian groups of toral automorphisms, Ergodic Theory Dyn. Syst., 15, 3, 569-592, (1995) · Zbl 0851.57039
[18] Marco, P. R., Non linearizable holomorphic dynamics having an uncountable number of symmetries, Inv. Math., 119, 67-127, (1995) · Zbl 0862.58045
[19] Marco, P. R., Total convergence or small divergence in small divisors, Commun. Math. Phys., 223, 3, 451-464, (2001) · Zbl 1161.37331
[20] Moser, J., On commuting circle mappings and simultaneous Diophantine approximations, Mathematische Zeitschrift, 205, 105-121, (1990) · Zbl 0689.58031
[21] Rousssarie, R., Modèles locaux de champs et de formes, 30, (1975), Astérisque · Zbl 0327.57017
[22] Schmidt, W., Modèles locaux de champs et de formes, 785, (1980), Springer Verlag
[23] Sternberg, S., The structure of local homeomorphisms II, III, Amer. J. Math., 623-632, (1958) · Zbl 0083.31406
[24] Stolovitch, L., Singular complete integrability, Publ. Math. I.H.E.S., 91, 134-210, (2000) · Zbl 0997.32024
[25] Stolovitch, L., Normalisation holomorphe d’algèbres de type Cartan de champs de vecteurs holomorphes singuliers, Ann. Math., 161, 589-612, (2005) · Zbl 1080.32019
[26] Walcher, S., On convergent normal form transformations in presence of symmetries, J. Math. Anal. Appl., 244, 17-26, (2000) · Zbl 0959.34030
[27] Yoccoz, J.-C, A remark on siegel’s theorem for nondiagonalizable linear part, Astérisque, 231, 3-88, (1995)
[28] Yoshino, M., Simultaneous normal forms of commuting maps and vector fields, World Scientific, Singapore, 287-294, (1999) · Zbl 0964.37029
[29] Zung, N. T., Convergence versus integrability in Poincaré-Dulac normal form., Math. Res. Lett., 9, 2-3, 217-228, (2002) · Zbl 1019.34084
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.