Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-05T04:15:56.437Z Has data issue: false hasContentIssue false

The Lyapunov exponents of generic zero divergence three-dimensional vector fields

Published online by Cambridge University Press:  01 October 2007

MÁRIO BESSA*
Affiliation:
IMPA, Estr. D. Castorina 110, 22460-320 Rio de Janeiro, Brazil (email: [email protected])

Abstract

We prove that for a C1-generic (dense Gδ) subset of all the conservative vector fields on three-dimensional compact manifolds without singularities, we have for Lebesgue almost every (a.e.) point pM that either the Lyapunov exponents at p are zero or X is an Anosov vector field. Then we prove that for a C1-dense subset of all the conservative vector fields on three-dimensional compact manifolds, we have for Lebesgue a.e. pM that either the Lyapunov exponents at p are zero or p belongs to a compact invariant set with dominated splitting for the linear Poincaré flow.

Type
Research Article
Copyright
Copyright © Cambridge University Press 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Ambrose, W. and Kakutani, S.. Structure and continuity of measure preserving transformations. Duke Math. J. 9 (1942), 2542.CrossRefGoogle Scholar
[2]Bessa, M.. The Lyapunov exponents of conservative continuous-time dynamical systems. Thesis, IMPA (C048/2006), 2005.Google Scholar
[3]Bessa, M.. Dynamics of generic two-dimensional linear differential systems. J. Differential Equations 228(2) (2006), 685706.CrossRefGoogle Scholar
[4]Bochi, J.. Genericity of zero Lyapunov exponents. Ergod. Th. & Dynam. Sys. 22 (2002), 16671696.CrossRefGoogle Scholar
[5]Bochi, J. and Viana, M.. The Lyapunov exponents of generic volume-preserving and symplectic maps. Ann. of Math. (2) 161 (2005), 14231485.CrossRefGoogle Scholar
[6]Bochi, J. and Viana, M.. Lyapunov exponents: How frequently are dynamical systems hyperbolic?. Advances in Dynamical Systems. Cambridge University Press, Cambridge, 2004.Google Scholar
[7]Dacorogna, B. and Moser, J.. On a partial differential equation involving the Jacobian determinant. Ann. Inst. H. Poincaré 7(1) (1990), 126.CrossRefGoogle Scholar
[8]Doering, C.. Persistently transitive vector fields on three-dimensional manifolds. Proc. Dynam. Sys. Bifur. Theory 160 (1987), 5989.Google Scholar
[9]Mañé, R.. Oseledec’s theorem from the generic viewpoint. Proc. Int. Congress of Mathematicians (Warszawa, 1983). Vol. 2. North-Holland, Amsterdam, 1984, pp. 12691276.Google Scholar
[10]Mañé, R.. The Lyapunov exponents of generic area preserving diffeomorphisms. Int. Conf. on Dynamical Systems (Montevideo, 1995) (Pitman Research Notes in Mathematics Series, 362). Longman, Harlow, 1996, pp. 110119.Google Scholar
[11]Moser, J.. On the volume elements on a manifold. Trans. Amer. Math. Soc. 120 (1965), 286294.CrossRefGoogle Scholar
[12]Oseledets, V. I.. A multiplicative ergodic theorem: Lyapunov characteristic numbers for dynamical systems. Trans. Moscow Math. Soc. 19 (1968), 197231.Google Scholar
[13]Robinson, C.. Generic properties of conservative systems. Amer. J. Math. 92 (1970), 562603.CrossRefGoogle Scholar
[14]Zuppa, C.. Regularisation des champs vectoriels qui préservent l’elément de volume. Bol. Soc. Brasil. Mat. 10(2) (1979), 5156.CrossRefGoogle Scholar