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

Published online by Cambridge University Press:  01 October 2007

MÁRIO BESSA
MÁRIO BESSA*
Affiliation:
IMPA, Estr. D. Castorina 110, 22460-320 Rio de Janeiro, Brazil (email: [email protected])
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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 p∈M 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. p∈M 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
Information
Ergodic Theory and Dynamical Systems , Volume 27 , Issue 5 , October 2007 , pp. 1445 - 1472
Copyright
Copyright © Cambridge University Press 2007

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References

[1]Ambrose, W. and Kakutani, S.. Structure and continuity of measure preserving transformations. Duke Math. J. 9 (1942), 25–42.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), 685–706.CrossRefGoogle Scholar
[4]Bochi, J.. Genericity of zero Lyapunov exponents. Ergod. Th. & Dynam. Sys. 22 (2002), 1667–1696.CrossRefGoogle Scholar
[5]Bochi, J. and Viana, M.. The Lyapunov exponents of generic volume-preserving and symplectic maps. Ann. of Math. (2) 161 (2005), 1423–1485.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), 1–26.CrossRefGoogle Scholar
[8]Doering, C.. Persistently transitive vector fields on three-dimensional manifolds. Proc. Dynam. Sys. Bifur. Theory 160 (1987), 59–89.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. 1269–1276.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. 110–119.Google Scholar
[11]Moser, J.. On the volume elements on a manifold. Trans. Amer. Math. Soc. 120 (1965), 286–294.CrossRefGoogle Scholar
[12]Oseledets, V. I.. A multiplicative ergodic theorem: Lyapunov characteristic numbers for dynamical systems. Trans. Moscow Math. Soc. 19 (1968), 197–231.Google Scholar
[13]Robinson, C.. Generic properties of conservative systems. Amer. J. Math. 92 (1970), 562–603.CrossRefGoogle Scholar
[14]Zuppa, C.. Regularisation des champs vectoriels qui préservent l’elément de volume. Bol. Soc. Brasil. Mat. 10(2) (1979), 51–56.CrossRefGoogle Scholar