Let f be a germ of biholomorphism of ℂ^n, fixing the origin. We show that if the germ commutes with a torus action, then we get information on the germs that can be conjugated to f, and furthermore on the existence of a holomorphic linearization or of a holomorphic normalization of f. We find out in a complete and computable manner what kind of structure a torus action must have in order to get a Poincaré-Dulac holomorphic normalization, studying the possible torsion phenomena. In particular, we link the eigenvalues of df_O to the weight matrix of the action. The link and the structure we found are more complicated than what one would expect; a detailed study was needed to completely understand the relations between torus actions, holomorphic Poincaré-Dulac normalizations, and torsion phenomena. We end the article giving an example of techniques that can be used to construct torus actions.

Raissy, J. (2010). Torus Actions in the Normalization Problem. THE JOURNAL OF GEOMETRIC ANALYSIS, 20(2), 472-524 [10.1007/s12220-009-9108-5].

Torus Actions in the Normalization Problem

RAISSY, JASMIN
2010

Abstract

Let f be a germ of biholomorphism of ℂ^n, fixing the origin. We show that if the germ commutes with a torus action, then we get information on the germs that can be conjugated to f, and furthermore on the existence of a holomorphic linearization or of a holomorphic normalization of f. We find out in a complete and computable manner what kind of structure a torus action must have in order to get a Poincaré-Dulac holomorphic normalization, studying the possible torsion phenomena. In particular, we link the eigenvalues of df_O to the weight matrix of the action. The link and the structure we found are more complicated than what one would expect; a detailed study was needed to completely understand the relations between torus actions, holomorphic Poincaré-Dulac normalizations, and torsion phenomena. We end the article giving an example of techniques that can be used to construct torus actions.
Articolo in rivista - Articolo scientifico
Normalization problem, Linearization problem, Resonances, Torus actions, Discrete local holomorphic dynamical systems
English
2010
20
2
472
524
none
Raissy, J. (2010). Torus Actions in the Normalization Problem. THE JOURNAL OF GEOMETRIC ANALYSIS, 20(2), 472-524 [10.1007/s12220-009-9108-5].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/9854
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