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Journal of Lie Theory 31 (2021), No. 3, 751--796
Copyright Heldermann Verlag 2021

Centralizers and Normalizers of Local Analytic and Formal Vector Fields

Niclas Kruff
Lehrstuhl A für Mathematik, RWTH Aachen, Germany

Sebastian Walcher
Lehrstuhl A für Mathematik, RWTH Aachen, Germany

Xiang Zhang
School of Mathematical Sciences and MOE-LSC, Shanghai Jiao Tong University, Shanghai, P. R. China

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.

Keywords: Local vector field, centralizer, normalizer, normal form, Jacobi multiplier.

MSC: 34A34, 34C14, 37G05, 37G40.

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