A generic incompressible flow is topological mixing

Mário Bessa

doi:10.1016/j.crma.2008.07.012

Dynamical Systems

A generic incompressible flow is topological mixing

Presented by: Étienne Ghys

Mário Bessa ¹

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, Volume 346 (2008) no. 21-22, pp. 1169-1174.

Abstract
Résumé

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 $τ \in R$ such that $X^{t} (U) \cap V \neq \emptyset$ for any $t ⩾ τ$ .

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 $X \in R$ est topologiquement mélangeant.

Received: 2008-04-16
Accepted: 2008-07-16
Published online: 2008-10-15

DOI: 10.1016/j.crma.2008.07.012

Author's affiliations:

Mário Bessa ¹

¹ ESTGOH – IPC, Rua General Santos Costa, 3400-124, Oliveira do Hospital and CMUP, Rua do Campo Alegre, 687, 4169-007 Porto, Portugal

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     title = {A generic incompressible flow is topological mixing},
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     pages = {1169--1174},
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     year = {2008},
     doi = {10.1016/j.crma.2008.07.012},
     language = {en},
}

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Mário Bessa. 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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2008.07.012/

References
Cited by

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

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

[3] M. Bessa 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 $C^{1}$ -robust transitivity of volume-preserving flows, J. Differential Equations (2008), in press

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

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

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

[8] C. Pugh; C. Robinson The $C^{1}$ closing lemma, including Hamiltonians, Ergodic Theory Dynam. Systems, Volume 3 (1983), pp. 261-313

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

[10] L. Wen; Z. Xia $C^{1}$ connecting lemmas, Trans. Amer. Math. Soc., Volume 352 (2000), pp. 5213-5230

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

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