Geodesic
Centralizers and Normalizers of Local Analytic and Formal Vector Fields
Journal of Lie theory, Tome 31 (2021) no. 3, pp. 751-796
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We investigate the structure of the centralizer and the normalizer of a local analytic or formal differential system at a nondegenerate stationary point, using the theory of Poincaré-Dulac normal forms. Our main results are concerned with the formal case. We obtain a description of the relation between centralizer and normalizer, sharp dimension estimates when the centralizer of the linearization has finite dimension, and lower estimates for the dimension of the centralizer in general. For a distinguished class of linear vector fields (which is sufficiently large to be of interest) we obtain a precise characterization of the centralizer for corresponding normal forms in the generic case. Moreover, in view of their relation to normalizers, we discuss inverse Jacobi multipliers and obtain existence criteria and nonexistence results for several classes of vector fields.
Classification : 34A34, 34C14, 37G05, 37G40
Mots-clés : Local vector field, centralizer, normalizer, normal form, Jacobi multiplier 
@article{JLT_2021_31_3_JLT_2021_31_3_a6,
     author = {N. Kruff and S. Walcher and X. Zhang},
     title = {Centralizers and {Normalizers} of {Local} {Analytic} and {Formal} {Vector} {Fields}},
     journal = {Journal of Lie theory},
     pages = {751--796},
     year = {2021},
     volume = {31},
     number = {3},
     url = {http://geodesic.mathdoc.fr/item/JLT_2021_31_3_JLT_2021_31_3_a6/}
}
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N. Kruff; S. Walcher; X. Zhang. Centralizers and Normalizers of Local Analytic and Formal Vector Fields. Journal of Lie theory, Tome 31 (2021) no. 3, pp. 751-796. http://geodesic.mathdoc.fr/item/JLT_2021_31_3_JLT_2021_31_3_a6/