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

DOI : https://doi.org/10.5802/aif.2453
Classification:  37F75,  32H02,  32A05,  40A05
Keywords: Holomorphic dynamical systems, diffeomorphisms, vector fields, potential theory
@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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {59},
     number = {3},
     year = {2009},
     pages = {951-975},
     doi = {10.5802/aif.2453},
     mrnumber = {2543658},
     zbl = {1186.37057},
     language = {en},
     url = {http://www.numdam.org/item/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. http://www.numdam.org/item/AIF_2009__59_3_951_0/

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