Simultaneous reduction to normal forms of commuting singular vector fields with linear parts having Jordan blocks
Annales de l'Institut Fourier, Volume 58 (2008) no. 1, pp. 263-297.

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.

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.

DOI: 10.5802/aif.2350
Classification: 32M25,  37F50,  37G05
Keywords: singular vector field, linearization, Jordan block, homological equation, Diophantine conditions, Gevrey spaces, decomposition
Yoshino, Masafumi 1; Gramchev, Todor 2

1 Graduate School of Science Hiroshima University Higashi-Hiroshima, 739-8526 (Japan)
2 Università di Cagliari Dipartimento di Matematica via Ospedale 72 09124 Cagliari (Italy)
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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, Volume 58 (2008) no. 1, pp. 263-297. doi : 10.5802/aif.2350. http://archive.numdam.org/articles/10.5802/aif.2350/

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

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

[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

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

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

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

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

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

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

[10] De La Llave, R. A tutorial on KAM theory. (1999) | Zbl

[11] DeLatte, D.; Gramchev, T. Biholomorphic maps with linear parts having Jordan blocks: Linearization and resonance type phenomena, Math. Physics Electronic Journal, Volume 8 (2002) no. paper n. 2, pp. 1-27 | MR | Zbl

[12] Dickinson, D.; Gramchev, T.; Yoshino, M. Perturbations of vector fields on tori: resonant normal forms and Diophantine phenomena, Proc. Edinb. Math. Soc. (2), Volume 45 (2002) no. 3, p. 731-159 | DOI | MR | Zbl

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

[14] Gantmacher, F. R. The theory of matrices, 1-2, Chelsea Publishing Co., New York, 1959 | MR | Zbl

[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

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

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

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

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

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

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

[22] Schmidt, W. Modèles locaux de champs et de formes, Lect. Notes in Mathematics, 785, Springer Verlag, 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) | DOI | MR | Zbl

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

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

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

[27] Yoccoz, J.-C A remark on Siegel’s theorem for nondiagonalizable linear part, Astérisque, Volume 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 | Zbl

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

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