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Javier Ribón
Non-embeddability of general unipotent diffeomorphisms up to formal conjugacy
(Non-existence de plongements de difféomorphismes unipotents à conjugaison formelle près)
Annales de l'institut Fourier, 59 no. 3 (2009), p. 951-975, doi: 10.5802/aif.2453
Article: subscription required (your ip address: 107.21.156.140) | Reviews MR 2543658 | Zbl 1186.37057
Class. Math.: 37F75, 32H02, 32A05, 40A05
Keywords: Holomorphic dynamical systems, diffeomorphisms, vector fields, potential theory
[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) no. 2, p. 543-572
Article | MR 1276933 | Zbl 0815.30018
[2] J. Ecalle, “Théorie des invariants holomorphes.”, Public. Math. Orsay (1974) no. 67
Article
[3] J. Ecalle, “Théorie itérative: introduction à la théorie des invariants holomorphes”, J. Math. Pures Appl. (9) 54 (1975), p. 183-258 MR 499882 | Zbl 0285.26010
[4] Yu. S. Ilyashenko, Nonlinear Stokes phenomena, Nonlinear Stokes phenomena, Amer. Math. Soc., 1993, p. 1–55 MR 1206039 | Zbl 0804.32011
[5] G. A. Kalyabin, “Asymptotics of the smallest eigenvalues of Hilbert-type matrices”, Funct. Anal. Appl. 35 (2001) no. 1, p. 67-70
Article | MR 1840752 | Zbl 1025.15013
[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), Birkhäuser, 1994, p. 96–116 MR 1279392 | Zbl 0797.58025
[7] B. Malgrange, Travaux d’Écalle et de Martinet-Ramis sur les systèmes dynamiques, Bourbaki Seminar, Vol. 1981/1982, Soc. Math. France, 1982, p. 59–73
Numdam | MR 689526 | Zbl 0526.58009
[8] J. Martinet & J.-P. Ramis, “Classification analytique des équations differentielles non linéaires résonnantes du premier ordre”, Ann. Sci. Ecole Norm. Sup. 4 (1983) no. 16, p. 571-621
Numdam | MR 740592 | Zbl 0534.34011
[9] J.-F. Mattei & R. Moussu, “Holonomie et intégrales premières”, Ann. Sci. École Norm. Sup. (4) 13 (1980) no. 4, p. 469-523
Numdam | MR 608290 | Zbl 0458.32005
[10] R. Pérez-Marco, “A note on holomorphic extensions”, Preprint. UCLA. http://xxx.lanl.gov/abs/math.DS/0009031, 2000
arXiv
[11] R. Pérez-Marco, “Total convergence or general divergence in small divisors”, Comm. Math. Phys. 223 (2001) no. 3, p. 451-464
Article | MR 1866162 | Zbl pre01731922
[12] R. Pérez-Marco, “Convergence or generic divergence of the Birkhoff normal form”, Ann. of Math. (2) 157 (2003) no. 2, p. 557-574
Article | MR 1973055 | Zbl 1038.37048
[13] T. Ransford, Potential theory in the complex plane, London Mathematical Society Student Texts 28, Cambridge University Press, 1995 MR 1334766 | Zbl 0828.31001
[14] J. Ribón, “Difféomorphismes de $(\mathbb{C}^{2},0)$ tangents à l’identité qui préservent la fibration de Hopf”, C. R. Acad. Sci. Paris Sér. I Math. 332 (2001) no. 11, p. 1011-1014 Zbl 1006.37024
[15] Javier Ribón, “Formal classification of unfoldings of parabolic diffeomorphisms”, Ergodic Theory Dynam. Systems 28 (2008) no. 4, p. 1323-1365
Article | MR 2437232 | Zbl 1153.37026
[16] J.-C. Tougeron, Idéaux de fonctions différentiables, Springer-Verlag, 1972, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 71 MR 440598 | Zbl 0251.58001
[17] S. M. Voronin, The Darboux-Whitney theorem and related questions, Nonlinear Stokes phenomena, Amer. Math. Soc., 1993, p. 139–233 MR 1206044 | Zbl 0789.58015
[18] S.M. Voronin, “Analytical classification of germs of conformal mappings $(\mathbb{C}^{},0) \rightarrow (\mathbb{C}^{},0)$ with identity linear part.”, Functional Anal. Appl. 1 (1981) no. 15, p. 1-13
Article | MR 609790 | Zbl 0463.30010
Javier Ribón
Non-embeddability of general unipotent diffeomorphisms up to formal conjugacy
(Non-existence de plongements de difféomorphismes unipotents à conjugaison formelle près)
Annales de l'institut Fourier, 59 no. 3 (2009), p. 951-975, doi: 10.5802/aif.2453
Article: subscription required (your ip address: 107.21.156.140) | Reviews MR 2543658 | Zbl 1186.37057
Class. Math.: 37F75, 32H02, 32A05, 40A05
Keywords: Holomorphic dynamical systems, diffeomorphisms, vector fields, potential theory
Résumé - 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 $\mathbb{C}^{n}$ ($n>1$) 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.
Bibliography
Article | MR 1276933 | Zbl 0815.30018
[2] J. Ecalle, “Théorie des invariants holomorphes.”, Public. Math. Orsay (1974) no. 67
Article
[3] J. Ecalle, “Théorie itérative: introduction à la théorie des invariants holomorphes”, J. Math. Pures Appl. (9) 54 (1975), p. 183-258 MR 499882 | Zbl 0285.26010
[4] Yu. S. Ilyashenko, Nonlinear Stokes phenomena, Nonlinear Stokes phenomena, Amer. Math. Soc., 1993, p. 1–55 MR 1206039 | Zbl 0804.32011
[5] G. A. Kalyabin, “Asymptotics of the smallest eigenvalues of Hilbert-type matrices”, Funct. Anal. Appl. 35 (2001) no. 1, p. 67-70
Article | MR 1840752 | Zbl 1025.15013
[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), Birkhäuser, 1994, p. 96–116 MR 1279392 | Zbl 0797.58025
[7] B. Malgrange, Travaux d’Écalle et de Martinet-Ramis sur les systèmes dynamiques, Bourbaki Seminar, Vol. 1981/1982, Soc. Math. France, 1982, p. 59–73
Numdam | MR 689526 | Zbl 0526.58009
[8] J. Martinet & J.-P. Ramis, “Classification analytique des équations differentielles non linéaires résonnantes du premier ordre”, Ann. Sci. Ecole Norm. Sup. 4 (1983) no. 16, p. 571-621
Numdam | MR 740592 | Zbl 0534.34011
[9] J.-F. Mattei & R. Moussu, “Holonomie et intégrales premières”, Ann. Sci. École Norm. Sup. (4) 13 (1980) no. 4, p. 469-523
Numdam | MR 608290 | Zbl 0458.32005
[10] R. Pérez-Marco, “A note on holomorphic extensions”, Preprint. UCLA. http://xxx.lanl.gov/abs/math.DS/0009031, 2000
arXiv
[11] R. Pérez-Marco, “Total convergence or general divergence in small divisors”, Comm. Math. Phys. 223 (2001) no. 3, p. 451-464
Article | MR 1866162 | Zbl pre01731922
[12] R. Pérez-Marco, “Convergence or generic divergence of the Birkhoff normal form”, Ann. of Math. (2) 157 (2003) no. 2, p. 557-574
Article | MR 1973055 | Zbl 1038.37048
[13] T. Ransford, Potential theory in the complex plane, London Mathematical Society Student Texts 28, Cambridge University Press, 1995 MR 1334766 | Zbl 0828.31001
[14] J. Ribón, “Difféomorphismes de $(\mathbb{C}^{2},0)$ tangents à l’identité qui préservent la fibration de Hopf”, C. R. Acad. Sci. Paris Sér. I Math. 332 (2001) no. 11, p. 1011-1014 Zbl 1006.37024
[15] Javier Ribón, “Formal classification of unfoldings of parabolic diffeomorphisms”, Ergodic Theory Dynam. Systems 28 (2008) no. 4, p. 1323-1365
Article | MR 2437232 | Zbl 1153.37026
[16] J.-C. Tougeron, Idéaux de fonctions différentiables, Springer-Verlag, 1972, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 71 MR 440598 | Zbl 0251.58001
[17] S. M. Voronin, The Darboux-Whitney theorem and related questions, Nonlinear Stokes phenomena, Amer. Math. Soc., 1993, p. 139–233 MR 1206044 | Zbl 0789.58015
[18] S.M. Voronin, “Analytical classification of germs of conformal mappings $(\mathbb{C}^{},0) \rightarrow (\mathbb{C}^{},0)$ with identity linear part.”, Functional Anal. Appl. 1 (1981) no. 15, p. 1-13
Article | MR 609790 | Zbl 0463.30010
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