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.
1
Lehrstuhl A für Mathematik, RWTH Aachen, Germany
2
School of Mathematical Sciences and MOE-LSC, Shanghai Jiao Tong University, Shanghai, P. R. China
Niclas Kruff; Sebastian Walcher; Xiang 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/JOLT_2021_31_3_a6/
@article{JOLT_2021_31_3_a6,
author = {Niclas Kruff and Sebastian Walcher and Xiang 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/JOLT_2021_31_3_a6/}
}
TY - JOUR
AU - Niclas Kruff
AU - Sebastian Walcher
AU - Xiang Zhang
TI - Centralizers and Normalizers of Local Analytic and Formal Vector Fields
JO - Journal of Lie Theory
PY - 2021
SP - 751
EP - 796
VL - 31
IS - 3
UR - http://geodesic.mathdoc.fr/item/JOLT_2021_31_3_a6/
ID - JOLT_2021_31_3_a6
ER -
%0 Journal Article
%A Niclas Kruff
%A Sebastian Walcher
%A Xiang Zhang
%T Centralizers and Normalizers of Local Analytic and Formal Vector Fields
%J Journal of Lie Theory
%D 2021
%P 751-796
%V 31
%N 3
%U http://geodesic.mathdoc.fr/item/JOLT_2021_31_3_a6/
%F JOLT_2021_31_3_a6
