Singularities of complex vector fields having many closed orbits
Journal of Singularities, Tome 8 (2014), pp. 69-82
Voir la notice de l'article provenant de la source Journal of Singularities website
A well-known result of Mattei and Moussu states that a germ of a holomorphic vector field at the origin in C^2 admits a holomorphic first integral if, and only if, the orbits are closed off the origin and only finitely many of these accumulate (only) at the origin. In this paper we investigate possible versions of such a result in terms of the measure of the set of closed orbits. We prove that if the set of closed leaves is a positive, i.e., a non-zero measure subset and the set of leaves accumulating only at the origin is a zero measure subset, then either there is a holomorphic first integral or the germ is formally linearizable as a suitable non-resonant singularity. The result is sharp as we show through some examples.
@article{10_5427_jsing_2014_8f,
author = {Bruno Sc\'ardua},
title = {Singularities of complex vector fields having many closed orbits},
journal = {Journal of Singularities},
pages = {69--82},
publisher = {mathdoc},
volume = {8},
year = {2014},
doi = {10.5427/jsing.2014.8f},
url = {http://geodesic.mathdoc.fr/articles/10.5427/jsing.2014.8f/}
}
Bruno Scárdua. Singularities of complex vector fields having many closed orbits. Journal of Singularities, Tome 8 (2014), pp. 69-82. doi: 10.5427/jsing.2014.8f
Cité par Sources :