no. 21-22
Dynamical Systems
A generic incompressible flow is topological mixing
[Un flot générique incompressible est mélangeant]

Présenté par : Étienne Ghys

Bessa, Mário1
1 ESTGOH – IPC, Rua General Santos Costa, 3400-124, Oliveira do Hospital and CMUP, Rua do Campo Alegre, 687, 4169-007 Porto, Portugal
Comptes Rendus. Mathématique, Tome 346 (2008) no. 21-22, pp. 1169-1174.

Dans cette Note nous montrons qu'il existe une partie résiduelle R dans l'ensemble des champs vectoriels qui préservent l'élément de volume pour laquelle tout X∈R est topologiquement mélangeant.

In this Note we prove that there exists a residual subset of the set of divergence-free vector fields defined on a compact, connected Riemannian manifold M, such that any vector field in this residual satisfies the following property: Given any two nonempty open subsets U and V of M, there exists τ∈R such that Xt(U)∩V≠∅ for any t⩾τ.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2008.07.012
Bessa, Mário 1

1 ESTGOH – IPC, Rua General Santos Costa, 3400-124, Oliveira do Hospital and CMUP, Rua do Campo Alegre, 687, 4169-007 Porto, Portugal 
@article{CRMATH_2008__346_21-22_1169_0,
     author = {Bessa, M\'ario},
     title = {A generic incompressible flow is topological mixing},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1169--1174},
     publisher = {Elsevier},
     volume = {346},
     number = {21-22},
     year = {2008},
     doi = {10.1016/j.crma.2008.07.012},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2008.07.012/}
}
TY  - JOUR
AU  - Bessa, Mário
TI  - A generic incompressible flow is topological mixing
JO  - Comptes Rendus. Mathématique
PY  - 2008
SP  - 1169
EP  - 1174
VL  - 346
IS  - 21-22
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2008.07.012/
DO  - 10.1016/j.crma.2008.07.012
LA  - en
ID  - CRMATH_2008__346_21-22_1169_0
ER  - 
%0 Journal Article
%A Bessa, Mário
%T A generic incompressible flow is topological mixing
%J Comptes Rendus. Mathématique
%D 2008
%P 1169-1174
%V 346
%N 21-22
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2008.07.012/
%R 10.1016/j.crma.2008.07.012
%G en
%F CRMATH_2008__346_21-22_1169_0
Bessa, Mário. A generic incompressible flow is topological mixing. Comptes Rendus. Mathématique, Tome 346 (2008) no. 21-22, pp. 1169-1174. doi : 10.1016/j.crma.2008.07.012. http://www.numdam.org/articles/10.1016/j.crma.2008.07.012/

[1] Abdenur, F.; Avila, A.; Bochi, J. Robust transitivity and topological mixing for C1-flows, Proc. Amer. Math. Soc., Volume 132 (2003) no. 3, pp. 699-705

[2] Arbieto, A.; Matheus, C. A pasting lemma and some applications for conservative systems, Ergodic Theory Dynam. Systems, Volume 27 (2007) no. 6, pp. 1399-1417

[3] Bessa, M. The Lyapunov exponents of generic zero divergence three-dimensional vector fields, Ergodic Theory Dynam. Systems, Volume 27 (2007) no. 6, pp. 1445-1472

[4] M. Bessa and J. Rocha, On C1-robust transitivity of volume-preserving flows, J. Differential Equations (2008), in press

[5] Bonatti, C.; Crovisier, S. Récurrence et généricité, Invent. Math., Volume 158 (2004) no. 1, pp. 33-104

[6] Moser, J. On the volume elements on a manifold, Trans. Amer. Math. Soc., Volume 120 (1965), pp. 286-294

[7] Palis, J.; de Melo, W. Geometric Theory of Dynamical Systems, Springer Verlag, 1982

[8] Pugh, C.; Robinson, C. The C1 closing lemma, including Hamiltonians, Ergodic Theory Dynam. Systems, Volume 3 (1983), pp. 261-313

[9] Robinson, C. Generic properties of conservative systems, Amer. J. Math., Volume 92 (1970), pp. 562-603

[10] Wen, L.; Xia, Z. C1 connecting lemmas, Trans. Amer. Math. Soc., Volume 352 (2000), pp. 5213-5230

[11] Zuppa, C. Regularisation C∞ des champs vectoriels qui préservent l'elément de volume, Bol. Soc. Bras. Mat., Volume 10 (1979) no. 2, pp. 51-56

Cité par Sources :