Non-embeddability of general unipotent diffeomorphisms up to formal conjugacy
Annales de l'Institut Fourier, Volume 59 (2009) no. 3, pp. 951-975.

The formal class of a germ of diffeomorphism ϕ is embeddable in a flow if ϕ is formally conjugated to the exponential of a germ of vector field. We prove that there are complex analytic unipotent germs of diffeomorphisms at n (n>1) whose formal class is non-embeddable. The examples are inside a family in which the non-embeddability is of geometrical type. The proof relies on the properties of some linear functional operators that we obtain through the study of polynomial families of diffeomorphisms via potential theory.

La classe formelle d’un difféomorphisme local ϕ est plongeable dans un flot si ϕ est formellement conjugué à l’exponentielle d’un germe de champs de vecteurs. On prouve qu’il existe des difféomorphismes unipotents analytiques complexes définis au voisinage de l’origine dans n (n>1) dont la classe formelle n’est pas plongeable. Les exemples appartiennent à une famille où le manque de plongeabilité est une propriété de type géométrique. La preuve est basée sur les propriétés de certains opérateurs fonctionnels linéaires qu’on obtient grâce à l’étude des familles polynomiales de difféomorphismes via la théorie du potentiel.

Received:
Revised:
Accepted:
DOI: 10.5802/aif.2453
Classification: 37F75,  32H02,  32A05,  40A05
Keywords: Holomorphic dynamical systems, diffeomorphisms, vector fields, potential theory
Ribón, Javier 1

1 UFF Instituto de Matemática Rua Mário Santos Braga S/N Valonguinho, Niterói, Rio de Janeiro 24020-14 (Brasil)
@article{AIF_2009__59_3_951_0,
     author = {Rib\'on, Javier},
     title = {Non-embeddability of general unipotent diffeomorphisms up to formal conjugacy},
     journal = {Annales de l'Institut Fourier},
     pages = {951--975},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {59},
     number = {3},
     year = {2009},
     doi = {10.5802/aif.2453},
     zbl = {1186.37057},
     mrnumber = {2543658},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2453/}
}
TY  - JOUR
TI  - Non-embeddability of general unipotent diffeomorphisms up to formal conjugacy
JO  - Annales de l'Institut Fourier
PY  - 2009
DA  - 2009///
SP  - 951
EP  - 975
VL  - 59
IS  - 3
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.2453/
UR  - https://zbmath.org/?q=an%3A1186.37057
UR  - https://www.ams.org/mathscinet-getitem?mr=2543658
UR  - https://doi.org/10.5802/aif.2453
DO  - 10.5802/aif.2453
LA  - en
ID  - AIF_2009__59_3_951_0
ER  - 
%0 Journal Article
%T Non-embeddability of general unipotent diffeomorphisms up to formal conjugacy
%J Annales de l'Institut Fourier
%D 2009
%P 951-975
%V 59
%N 3
%I Association des Annales de l’institut Fourier
%U https://doi.org/10.5802/aif.2453
%R 10.5802/aif.2453
%G en
%F AIF_2009__59_3_951_0
Ribón, Javier. Non-embeddability of general unipotent diffeomorphisms up to formal conjugacy. Annales de l'Institut Fourier, Volume 59 (2009) no. 3, pp. 951-975. doi : 10.5802/aif.2453. https://aif.centre-mersenne.org/articles/10.5802/aif.2453/

[1] Ahern, Patrick; Rosay, Jean-Pierre Entire functions, in the classification of differentiable germs tangent to the identity, in one or two variables, Trans. Amer. Math. Soc., Volume 347 (1995) no. 2, pp. 543-572 | Article | MR: 1276933 | Zbl: 0815.30018

[2] Ecalle, J. Théorie des invariants holomorphes., Public. Math. Orsay (1974) no. 67, 206 pages

[3] Ecalle, J. Théorie itérative: introduction à la théorie des invariants holomorphes, J. Math. Pures Appl. (9), Volume 54 (1975), pp. 183-258 | MR: 499882 | Zbl: 0285.26010

[4] Ilyashenko, Yu. S. Nonlinear Stokes phenomena, Nonlinear Stokes phenomena (Adv. Soviet Math.) Volume 14, Amer. Math. Soc., Providence, RI, 1993, pp. 1-55 | MR: 1206039 | Zbl: 0804.32011

[5] Kalyabin, G. A. Asymptotics of the smallest eigenvalues of Hilbert-type matrices, Funct. Anal. Appl., Volume 35 (2001) no. 1, pp. 67-70 | Article | MR: 1840752 | Zbl: 1025.15013

[6] Kuksin, S.; Pöschel, J. On the inclusion of analytic symplectic maps in analytic Hamiltonian flows and its applications, Seminar on Dynamical Systems (St. Petersburg, 1991) (Progr. Nonlinear Differential Equations Appl.) Volume 12, Birkhäuser, Basel, 1994, pp. 96-116 | MR: 1279392 | Zbl: 0797.58025

[7] Malgrange, B. Travaux d’Écalle et de Martinet-Ramis sur les systèmes dynamiques, Bourbaki Seminar, Vol. 1981/1982 (Astérisque) Volume 92, Soc. Math. France, Paris, 1982, pp. 59-73 | Numdam | MR: 689526 | Zbl: 0526.58009

[8] Martinet, J.; Ramis, J.-P. Classification analytique des équations differentielles non linéaires résonnantes du premier ordre, Ann. Sci. Ecole Norm. Sup., Volume 4 (1983) no. 16, pp. 571-621 | Numdam | MR: 740592 | Zbl: 0534.34011

[9] Mattei, J.-F.; Moussu, R. Holonomie et intégrales premières, Ann. Sci. École Norm. Sup. (4), Volume 13 (1980) no. 4, pp. 469-523 | Numdam | MR: 608290 | Zbl: 0458.32005

[10] Pérez-Marco, R. A note on holomorphic extensions (2000) (Preprint. UCLA. http://xxx.lanl.gov/abs/math.DS/0009031)

[11] Pérez-Marco, R. Total convergence or general divergence in small divisors, Comm. Math. Phys., Volume 223 (2001) no. 3, pp. 451-464 | Article | MR: 1866162 | Zbl: pre01731922

[12] Pérez-Marco, R. Convergence or generic divergence of the Birkhoff normal form, Ann. of Math. (2), Volume 157 (2003) no. 2, pp. 557-574 | Article | MR: 1973055 | Zbl: 1038.37048

[13] Ransford, T. Potential theory in the complex plane, London Mathematical Society Student Texts, Volume 28, Cambridge University Press, Cambridge, 1995 | MR: 1334766 | Zbl: 0828.31001

[14] Ribón, J. Difféomorphismes de ( 2 ,0) tangents à l’identité qui préservent la fibration de Hopf, C. R. Acad. Sci. Paris Sér. I Math., Volume 332 (2001) no. 11, pp. 1011-1014 | MR: 1838129 | Zbl: 1006.37024

[15] Ribón, Javier Formal classification of unfoldings of parabolic diffeomorphisms, Ergodic Theory Dynam. Systems, Volume 28 (2008) no. 4, pp. 1323-1365 | Article | MR: 2437232 | Zbl: 1153.37026

[16] Tougeron, J.-C. Idéaux de fonctions différentiables, Springer-Verlag, Berlin, 1972 (Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 71) | MR: 440598 | Zbl: 0251.58001

[17] Voronin, S. M. The Darboux-Whitney theorem and related questions, Nonlinear Stokes phenomena (Adv. Soviet Math.) Volume 14, Amer. Math. Soc., Providence, RI, 1993, pp. 139-233 | MR: 1206044 | Zbl: 0789.58015

[18] Voronin, S.M. Analytical classification of germs of conformal mappings ( ,0)( ,0) with identity linear part., Functional Anal. Appl., Volume 1 (1981) no. 15, pp. 1-13 | Article | MR: 609790 | Zbl: 0463.30010

Cited by Sources: