@ARTICLE{10.21494/ISTE.OP.2022.0837, 

TITLE={Limit sets and global dynamic for 2-D divergence-free vector fields}, 
  
AUTHOR={Habib Marzougui, }, 
	
JOURNAL={Advances in Pure and Applied Mathematics}, 
	
VOLUME={13}, 

NUMBER={Issue 3 (June 2022)}, 
	
YEAR={2022}, 

URL={https://openscience.fr/Limit-sets-and-global-dynamic-for-2-d-divergence-free-vector-fields}, 
	
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.}}