Exponentially long time stability for non-linearizable analytic germs of ( n ,0).  [ Temps de stabilité exponentiellement longs pour les germes analytiques de ( n ,0) non linéarisables. ]
Annales de l'Institut Fourier, Tome 54 (2004) no. 4, pp. 989-1004.

Nous étudions le problème du centre de Siegel-Schröder, sur la linéarisation de germes analytiques de plusieurs variables complexes, dans la catégorie Gevrey-s. Nous introduisons une nouvelle condition arithmétique de type de Bruno, sur la partie linéaire du germe, qui assure l’existence d’une linéarisation formelle Gevrey-s. Nous l’utilisons pour démontrer la stabilité effective, c’est-à-dire stabilité pour un temps fini mais long, d’un voisinage du point fixe, pour le germe analytique.

We study the Siegel-Schröder center problem on the linearization of analytic germs of diffeomorphisms in several complex variables, in the Gevrey-s, s>0 category. We introduce a new arithmetical condition of Bruno type on the linear part of the given germ, which ensures the existence of a Gevrey-s formal linearization. We use this fact to prove the effective stability, i.e. stability for finite but long time, of neighborhoods of the origin, for the analytic germ.

DOI : https://doi.org/10.5802/aif.2040
Classification : 37F50,  70H14
Mots clés : problème du centre de Siegel, classe Gevrey, condition de Bruno, stabilité effective, estimations type Nekoroshev
@article{AIF_2004__54_4_989_0,
     author = {Carletti, Timoteo},
     title = {Exponentially long time stability for non-linearizable analytic germs of $({\mathbb {C}}^n,0)$.},
     journal = {Annales de l'Institut Fourier},
     pages = {989--1004},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {54},
     number = {4},
     year = {2004},
     doi = {10.5802/aif.2040},
     zbl = {1063.37043},
     mrnumber = {2111018},
     language = {en},
     url = {archive.numdam.org/item/AIF_2004__54_4_989_0/}
}
Carletti, Timoteo. Exponentially long time stability for non-linearizable analytic germs of $({\mathbb {C}}^n,0)$.. Annales de l'Institut Fourier, Tome 54 (2004) no. 4, pp. 989-1004. doi : 10.5802/aif.2040. http://archive.numdam.org/item/AIF_2004__54_4_989_0/

[Ba] W. Balser From Divergent Power Series to Analytic Functions. Theory and Applications of Multisummable Power Series, Lectures Notes in Mathematics, Volume 1582, Springer, 1994 | MR 1317343 | Zbl 0810.34046

[Br] A.D. Bruno Analytical form of differential equations, Transactions Moscow Math.Soc, Volume 25 (1971), pp. 131-288 | Zbl 0272.34018

[Br] A.D. Bruno Analytical form of differential equations, Transactions Moscow Math. Soc., Volume 26 (1972), pp. 199-239 | Zbl 0269.34006

[Ca] T. Carletti The Lagrange inversion formula on non--Archimedean fields. Non--Analytical Form of Differential and Finite Difference Equations, DCDS Séries A, Volume 9 (2003) no. 4, pp. 835-858 | MR 1903046 | Zbl 1036.37017

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

[GDFGS] A. Giorgilli; A. Fontich; L. Galgani; C. Simó Effective stability for a Hamiltonian system near an elliptic equilibrium point with an application to the restricted three body problem, J. of Differential Equations, Volume 77 (1989), pp. 167-198 | MR 980547 | Zbl 0675.70027

[Gr] A. Gray A fixed point theorem for small divisors problems, J. Diff. Eq, Volume 18 (1975), pp. 346-365 | MR 375389 | Zbl 0301.58014

[He] M.R. Herman; Mebkhout Seneor Eds. Recent Results and Some Open Questions on Siegel's Linearization Theorem of Germs of Complex Analytic Diffeomorphisms of n near a Fixed Point, Proc. VIII Int. Conf. Math. Phys. (1986), pp. 138-184

[HW] G.H. Hardy; E.M. Wright An introduction to the theory of numbers, Oxford Univ. Press | MR 67125 | Zbl 0058.03301

[Ko] G. Koenigs Recherches sur les équations fonctionelles, Ann. Sc. E.N.S., Volume 1 (1884) no. supplément, pp. 3-41 | JFM 16.0376.01 | Numdam | MR 1508749

[MMY] S. Marmi; P. Moussa; J.-C. Yoccoz The Bruno functions and their regularity properties, Communications in Mathematical Physics, Volume 186 (1997), pp. 265-293 | MR 1462766 | Zbl 0947.30018

[Ne] N. N. Nekhoroshev An exponential estimate of the time of stability of nearly integrable Hamiltonian systems, Usp. Math. Nauk, Volume 32 (1977), pp. 5-66 | MR 501140 | Zbl 0389.70028

[Ne] N.N. Nekhoroshev An exponential estimate of the time of stability of nearly-integrable Hamiltonian systems., Russ. Math. Surv., Volume 32 (1977) no. 6, pp. 1-65 | MR 501140 | Zbl 0389.70028

[PM2] R. Pérez--Marco Sur la dynamique des germes de difféomorphismes de ( , 0 ) et des difféomorphismes analytiques du cercle (1990) (Thèse Université de Paris Sud)

[PM1] R. Pérez--Marco Sur les dynamiques holomorphes non linéarisables et une conjecture de V.I. Arnold, Ann. scient. Éc. Norm. Sup. (4), Volume 26 (1993), pp. 565-644 | Numdam | MR 1241470 | Zbl 0812.58051

[Po] H. Poincaré Œuvres Volume tome I, Gauthier--Villars, Paris, 1917

[Ra] J.--P. Ramis Séries divergentes et Théorie asymptotiques, Publ. Journées X--UPS (1991), pp. 1-67

[Si] C.L. Siegel Iteration of analytic functions, Annals of Mathematics, Volume 43 (1942), pp. 807-812 | MR 7044 | Zbl 0061.14904

[St] S. Sternberg Infinite Lie groups and the formal aspects of dynamical systems, J. Math. Mech, Volume 10 (1961), pp. 451-474 | MR 133400 | Zbl 0131.26802

[Yo] J.-C. Yoccoz Théorème de Siegel, polynômes quadratiques et nombres de Bruno, Astérisque, Volume 231 (1995), pp. 3-88 | MR 1367353