Simultaneous reduction to normal forms of commuting singular vector fields with linear parts having Jordan blocks  [ Réduction simultanée aux formes normales de champs de vecteurs singuliers commutatifs avec des parties linéaires ayant des blocs de Jordan ]
Annales de l'Institut Fourier, Tome 58 (2008) no. 1, p. 263-297
Nous étudions la linéarisation simultanée de d–actions (et les algèbres correspondants de Lie d–dimensionelles) definie par des champs de vecteurs singuliers dans n fixant l’origine avec des parties linéaires ayant des blocs de Jordan. Nous montrons la convergence analytique des transformations linéarisantes formelles sous une condition d’invariance géométrique pour le spectre de d-champs de vecteurs qui engendrent une algèbre de Lie. Si la condition n’est pas satisfaite et si il y a des petits diviseurs, nous montrons l’existence de solutions divergentes pour un système sous déterminé d’équations linéarisées homologiques. Dans le cadre de fonctions la situation est complètement différente. Nous montrons le théorème de Sternberg pour une famille commutative de champs de vecteurs qui ne satisfait pas la condition.
We study the simultaneous linearizability of d–actions (and the corresponding d-dimensional Lie algebras) defined by commuting singular vector fields in 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 category, the situation is completely different. We show Sternberg’s theorem for a commuting system of vector fields with a Jordan block although they do not satisfy the condition.
DOI : https://doi.org/10.5802/aif.2350
Classification:  32M25,  37F50,  37G05
Mots clés: champ de vecteurs singulier, linéarisation, bloc de Jordan, équations omologiques, conditions diophantiennes, espaces de Gevrey, décomposition
@article{AIF_2008__58_1_263_0,
     author = {Yoshino, Masafumi and Gramchev, Todor},
     title = {Simultaneous reduction to normal forms of commuting singular vector fields with linear parts having Jordan blocks},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {58},
     number = {1},
     year = {2008},
     pages = {263-297},
     doi = {10.5802/aif.2350},
     mrnumber = {2401222},
     zbl = {1137.37025},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2008__58_1_263_0}
}
Yoshino, Masafumi; Gramchev, Todor. Simultaneous reduction to normal forms of commuting singular vector fields with linear parts having Jordan blocks. Annales de l'Institut Fourier, Tome 58 (2008) no. 1, pp. 263-297. doi : 10.5802/aif.2350. http://www.numdam.org/item/AIF_2008__58_1_263_0/

[1] Abate, M. Diagonalization of nondiagonalizable discrete holomorphic dynamical systems, Amer. J. Math., Tome 122 (2000), pp. 757-781 | Article | MR 1771573 | Zbl 0966.32018

[2] Arnold, V. I. Geometrical Methods in the Theory of Ordinary Differential Equations, Springer (1983) | MR 695786 | Zbl 0507.34003

[3] Bruno, A. D. The analytic form of differential equations, Tr. Mosk. Mat. O-va (1971) no. 25, pp. 119-262 (and 26, 199–239 (1972) (in Russian); see also Trans. Mosc. Math. Soc. 25, 131-288 (1971) and 26, 199-239 (1972)) | Zbl 0263.34003

[4] Bruno, A. D.; Walcher, S. Symmetries and convergence of normalizing transformations, J. Math. Anal. Appl., Tome 183 (1994), pp. 571-576 | Article | MR 1274857 | Zbl 0804.34040

[5] Carletti, T. Exponentially long time stability for non-linearizable analytic germs of ( n ,0), Ann Inst. Fourier (Grenoble), Tome 54 (2004) no. 4, pp. 989-1004 | Article | Numdam | MR 2111018 | Zbl 1063.37043

[6] Carletti, T.; Marmi, S. Linearization of analytic and non-analytic germs of diffeomorphisms of (,0), Bull. Soc. Math. France, Tome 128 (2000), pp. 69-85 | Numdam | MR 1765828 | Zbl 0997.37017

[7] Chen, K. T. Diffeomorphisms: C -realizations of formal properties, Amer. J. Math., Tome 87 (1965), pp. 140-157 | Article | MR 173271 | Zbl 0151.32001

[8] Cicogna, G.; Gaeta, G. Symmetry and perturbation theory in nonlinear dynamics, Springer–Verlag, New Series m: Monographs, Tome 57 (1999) | MR 1732242 | Zbl 1059.37044

[9] Cicogna, G.; Walcher, S. Convergence of normal form transformations: the role of symmetries. Symmetry and perturbation theory, Acta Math. Appl., Tome 70 (2002), pp. 95-111 | Article | MR 1892377 | Zbl 1013.34033

[10] De La Llave, R. A tutorial on KAM theory., Univ. of Washington, Seattle, to appear in Proc. of 1999 Summer Research Institute on Smooth Ergodic Theory and Applications (1999) | 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, Tome 8 (2002) no. paper n. 2, pp. 1-27 | MR 1922424 | 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), Tome 45 (2002) no. 3, p. 731-159 | Article | MR 1933753 | Zbl 1032.37010

[13] Dumortier, F.; Roussarie, R. Smooth linearization of germs of R 2 -actions and holomorphic vector fields, Ann. Inst. Fourier (Grenoble), Tome 30 (1980) no. 1, pp. 31-64 | Article | Numdam | MR 576072 | Zbl 0418.58015

[14] Gantmacher, F. R. The theory of matrices, Chelsea Publishing Co., New York Tome 1-2 (1959) | MR 107649 | 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 (2003), pp. 106-115 (S. Abenda, G. Gaeta and S. Walcher eds, Symmetry and perturbation theory, Cala Gonone, 16–22 May 2002) | MR 1976662

[16] Herman, M. Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Publ. Math. I.H.É.S., Tome 49 (1979), pp. 5-233 | Numdam | MR 538680 | Zbl 0448.58019

[17] Katok, A.; Katok, S. Higher cohomology for Abelian groups of toral automorphisms, Ergodic Theory Dyn. Syst., Tome 15 (1995) no. 3, pp. 569-592 | Article | MR 1336707 | Zbl 0851.57039

[18] Marco, P. R. Non linearizable holomorphic dynamics having an uncountable number of symmetries, Inv. Math., Tome 119 (1995), pp. 67-127 | Article | MR 1309972 | Zbl 0862.58045

[19] Marco, P. R. Total convergence or small divergence in small divisors, Commun. Math. Phys., Tome 223 (2001) no. 3, pp. 451-464 | Article | MR 1866162 | Zbl 01731922

[20] Moser, J. On commuting circle mappings and simultaneous Diophantine approximations, Mathematische Zeitschrift, Tome 205 (1990), pp. 105-121 | Article | MR 1069487 | Zbl 0689.58031

[21] Rousssarie, R. Modèles locaux de champs et de formes, Astérisque Tome 30 (1975) | Zbl 0327.57017

[22] Schmidt, W. Modèles locaux de champs et de formes, Springer Verlag, Lect. Notes in Mathematics, Tome 785 (1980)

[23] Sternberg, S. The structure of local homeomorphisms II, III, Amer. J. Math. (1958), pp. 623-632 (80, 623-632 and 81, 578–604) | Article | MR 96854 | Zbl 0083.31406

[24] Stolovitch, L. Singular complete integrability, Publ. Math. I.H.E.S., Tome 91 (2000), pp. 134-210 | Numdam | MR 1828744 | Zbl 0997.32024

[25] Stolovitch, L. Normalisation holomorphe d’algèbres de type Cartan de champs de vecteurs holomorphes singuliers, Ann. Math., Tome 161 (2005), pp. 589-612 | Article | Zbl 1080.32019

[26] Walcher, S. On convergent normal form transformations in presence of symmetries, J. Math. Anal. Appl., Tome 244 (2000), pp. 17-26 | Article | MR 1746785 | Zbl 0959.34030

[27] Yoccoz, J.-C A remark on Siegel’s theorem for nondiagonalizable linear part, Astérisque, Tome 231 (1995), pp. 3-88 (manuscript, 1978, see also Théorème de Siegel, nombres de Bruno et polynômes quadratiques)

[28] Yoshino, M. Simultaneous normal forms of commuting maps and vector fields, World Scientific, Singapore (1999), pp. 287-294 (A. Degasperis, G. Gaeta eds., Symmetry and perturbation theory SPT 98, Rome 16–22 December 1998) | MR 1844128 | Zbl 0964.37029

[29] Zung, N. T. Convergence versus integrability in Poincaré-Dulac normal form., Math. Res. Lett., Tome 9 (2002) no. 2-3, pp. 217-228 | MR 1909639 | Zbl 1019.34084