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 such that for any .
Dans cette Note nous montrons qu'il existe une partie résiduelle dans l'ensemble des champs vectoriels qui préservent l'élément de volume pour laquelle tout est topologiquement mélangeant.
Accepted:
Published online:
@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://archive.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://archive.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://archive.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, Volume 346 (2008) no. 21-22, pp. 1169-1174. doi : 10.1016/j.crma.2008.07.012. http://archive.numdam.org/articles/10.1016/j.crma.2008.07.012/
[1] Robust transitivity and topological mixing for -flows, Proc. Amer. Math. Soc., Volume 132 (2003) no. 3, pp. 699-705
[2] A pasting lemma and some applications for conservative systems, Ergodic Theory Dynam. Systems, Volume 27 (2007) no. 6, pp. 1399-1417
[3] 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 -robust transitivity of volume-preserving flows, J. Differential Equations (2008), in press
[5] Récurrence et généricité, Invent. Math., Volume 158 (2004) no. 1, pp. 33-104
[6] On the volume elements on a manifold, Trans. Amer. Math. Soc., Volume 120 (1965), pp. 286-294
[7] Geometric Theory of Dynamical Systems, Springer Verlag, 1982
[8] The closing lemma, including Hamiltonians, Ergodic Theory Dynam. Systems, Volume 3 (1983), pp. 261-313
[9] Generic properties of conservative systems, Amer. J. Math., Volume 92 (1970), pp. 562-603
[10] connecting lemmas, Trans. Amer. Math. Soc., Volume 352 (2000), pp. 5213-5230
[11] Regularisation des champs vectoriels qui préservent l'elément de volume, Bol. Soc. Bras. Mat., Volume 10 (1979) no. 2, pp. 51-56
Cited by Sources: