@ARTICLE{10.21494/ISTE.OP.2022.0837, 

TITLE={Ensembles limites et dynamique globale pour les 2-D champs de vecteurs sans divergence}, 
  
AUTHOR={Habib Marzougui, }, 
	
JOURNAL={Avancées en Mathématiques Pures et Appliquées}, 
	
VOLUME={13}, 

NUMBER={Numéro 3 (Juin 2022)}, 
	
YEAR={2022}, 

URL={https://www.openscience.fr/Ensembles-limites-et-dynamique-globale-pour-les-2-D-champs-de-vecteurs-sans}, 
	
DOI={10.21494/ISTE.OP.2022.0837}, 
	
ISSN={1869-6090}, 
	
ABSTRACT={The global structure of divergence-free vector fields on closed surfaces is investigated. We prove that if M is a closed surface and $$${V}$$$ is a divergence-free $$$C^{1}$$$-vector field with finitely many singularities on M then every orbit L of $$$\mathcal{V}$$$ is one of the following types : (i) a singular point, (ii) a periodic orbit, (iii) a closed (non periodic) orbit in M* = M - Sing($$$\mathcal{V}$$$), (iv) a locally dense orbit, where Sing($$$\mathcal{V}$$$) denotes the set of singular points of $$$\mathcal{V}$$$. On the other hand, we show that the complementary in M of periodic components and minimal components is a compact invariant subset consisting of singularities and closed (non compact) orbits in M*. These results extend those of T. Ma and S. Wang in [Discrete Contin. Dynam. Systems, 7 (2001), 431-445] established when the divergence-free vector field $$$\mathcal{V}$$$ is regular that is all its singular points are non-degenerate.}}