Non-embeddability of general unipotent diffeomorphisms up to formal conjugacy

Javier Ribón[1]

  • [1] UFF Instituto de Matemática Rua Mário Santos Braga S/N Valonguinho, Niterói, Rio de Janeiro 24020-14 (Brasil)

Annales de l’institut Fourier (2009)

  • Volume: 59, Issue: 3, page 951-975
  • ISSN: 0373-0956

Abstract

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The formal class of a germ of diffeomorphism is embeddable in a flow if is formally conjugated to the exponential of a germ of vector field. We prove that there are complex analytic unipotent germs of diffeomorphisms at () 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.

How to cite

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Ribón, Javier. "Non-embeddability of general unipotent diffeomorphisms up to formal conjugacy." Annales de l’institut Fourier 59.3 (2009): 951-975. <http://eudml.org/doc/10425>.

@article{Ribón2009,
abstract = {The formal class of a germ of diffeomorphism $\varphi $ is embeddable in a flow if $\varphi $ is formally conjugated to the exponential of a germ of vector field. We prove that there are complex analytic unipotent germs of diffeomorphisms at $\mathbb\{C\}^\{n\}$ ($n&gt;1$) 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.},
affiliation = {UFF Instituto de Matemática Rua Mário Santos Braga S/N Valonguinho, Niterói, Rio de Janeiro 24020-14 (Brasil)},
author = {Ribón, Javier},
journal = {Annales de l’institut Fourier},
keywords = {Holomorphic dynamical systems; diffeomorphisms; vector fields; potential theory; holomorphic dynamical systems; infinitesimal generator; exponential map},
language = {eng},
number = {3},
pages = {951-975},
publisher = {Association des Annales de l’institut Fourier},
title = {Non-embeddability of general unipotent diffeomorphisms up to formal conjugacy},
url = {http://eudml.org/doc/10425},
volume = {59},
year = {2009},
}

TY - JOUR
AU - Ribón, Javier
TI - Non-embeddability of general unipotent diffeomorphisms up to formal conjugacy
JO - Annales de l’institut Fourier
PY - 2009
PB - Association des Annales de l’institut Fourier
VL - 59
IS - 3
SP - 951
EP - 975
AB - The formal class of a germ of diffeomorphism $\varphi $ is embeddable in a flow if $\varphi $ is formally conjugated to the exponential of a germ of vector field. We prove that there are complex analytic unipotent germs of diffeomorphisms at $\mathbb{C}^{n}$ ($n&gt;1$) 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 - eng
KW - Holomorphic dynamical systems; diffeomorphisms; vector fields; potential theory; holomorphic dynamical systems; infinitesimal generator; exponential map
UR - http://eudml.org/doc/10425
ER -

References

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  1. Patrick Ahern, Jean-Pierre Rosay, Entire functions, in the classification of differentiable germs tangent to the identity, in one or two variables, Trans. Amer. Math. Soc. 347 (1995), 543-572 Zbl0815.30018MR1276933
  2. J. Ecalle, Théorie des invariants holomorphes., Public. Math. Orsay (1974) Zbl0285.26010
  3. J. Ecalle, Théorie itérative: introduction à la théorie des invariants holomorphes, J. Math. Pures Appl. (9) 54 (1975), 183-258 Zbl0285.26010MR499882
  4. Yu. S. Ilyashenko, Nonlinear Stokes phenomena, Nonlinear Stokes phenomena 14 (1993), 1-55, Amer. Math. Soc., Providence, RI Zbl0804.32011MR1206039
  5. G. A. Kalyabin, Asymptotics of the smallest eigenvalues of Hilbert-type matrices, Funct. Anal. Appl. 35 (2001), 67-70 Zbl1025.15013MR1840752
  6. S. Kuksin, J. Pöschel, On the inclusion of analytic symplectic maps in analytic Hamiltonian flows and its applications, Seminar on Dynamical Systems (St. Petersburg, 1991) 12 (1994), 96-116, Birkhäuser, Basel Zbl0797.58025MR1279392
  7. B. Malgrange, Travaux d’Écalle et de Martinet-Ramis sur les systèmes dynamiques, Bourbaki Seminar, Vol. 1981/1982 92 (1982), 59-73, Soc. Math. France, Paris Zbl0526.58009MR689526
  8. J. Martinet, J.-P. Ramis, Classification analytique des équations differentielles non linéaires résonnantes du premier ordre, Ann. Sci. Ecole Norm. Sup. 4 (1983), 571-621 Zbl0534.34011MR740592
  9. J.-F. Mattei, R. Moussu, Holonomie et intégrales premières, Ann. Sci. École Norm. Sup. (4) 13 (1980), 469-523 Zbl0458.32005MR608290
  10. R. Pérez-Marco, A note on holomorphic extensions, (2000) 
  11. R. Pérez-Marco, Total convergence or general divergence in small divisors, Comm. Math. Phys. 223 (2001), 451-464 Zbl1161.37331MR1866162
  12. R. Pérez-Marco, Convergence or generic divergence of the Birkhoff normal form, Ann. of Math. (2) 157 (2003), 557-574 Zbl1038.37048MR1973055
  13. T. Ransford, Potential theory in the complex plane, 28 (1995), Cambridge University Press, Cambridge Zbl0828.31001MR1334766
  14. J. Ribón, Difféomorphismes de tangents à l’identité qui préservent la fibration de Hopf, C. R. Acad. Sci. Paris Sér. I Math. 332 (2001), 1011-1014 Zbl1006.37024
  15. Javier Ribón, Formal classification of unfoldings of parabolic diffeomorphisms, Ergodic Theory Dynam. Systems 28 (2008), 1323-1365 Zbl1153.37026MR2437232
  16. J.-C. Tougeron, Idéaux de fonctions différentiables, (1972), Springer-Verlag, Berlin Zbl0251.58001MR440598
  17. S. M. Voronin, The Darboux-Whitney theorem and related questions, Nonlinear Stokes phenomena 14 (1993), 139-233, Amer. Math. Soc., Providence, RI Zbl0789.58015MR1206044
  18. S.M. Voronin, Analytical classification of germs of conformal mappings with identity linear part., Functional Anal. Appl. 1 (1981), 1-13 Zbl0463.30010MR609790

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