论文信息 - THREE-DIMENSIONAL CONSERVATIVE STAR FLOWS ARE ANOSOV

THREE-DIMENSIONAL CONSERVATIVE STAR FLOWS ARE ANOSOV

A divergence-free vector field satisfies the star property if any divergence-free vector field in some $C^1$-neighborhood has all the singularities and all closed orbits hyperbolic. In this article we prove that any divergence-free star vector field defined in a closed three-dimensional manifold is Anosov. Moreover, we prove that a $C^1$-structurally stable three-dimensional conservative flow is Anosov.

Jorge Rocha | Mário Bessa | J. Rocha | M. Bessa

[1] A. Arbieto,et al. A pasting lemma and some applications for conservative systems , 2006, Ergodic Theory and Dynamical Systems.

[2] V. Araújo,et al. Dominated splitting and zero volume for incompressible three flows , 2008, 0801.2148.

[3] P. Duarte,et al. Abundance of elliptic dynamics on conservative three-flows , 2008 .

[4] Thérèse Vivier. Projective hyperbolicity and fixed points , 2006, Ergodic Theory and Dynamical Systems.

[5] Charles Pugh,et al. The C1 Closing Lemma, including Hamiltonians , 1983, Ergodic Theory and Dynamical Systems.

[6] Shaobo Gan,et al. Nonsingular star flows satisfy Axiom A and the no-cycle condition , 2006 .

[7] J. Bochi. Genericity of zero Lyapunov exponents , 2002, Ergodic Theory and Dynamical Systems.

[8] Carlos Zuppa. Regularisation C∞ des champs vectoriels qui préservent l'élément de volume , 1979 .

[9] J. Palis,et al. Geometric theory of dynamical systems , 1982 .

[10] J. Rocha,et al. On C1-robust transitivity of volume-preserving flows , 2007, 0707.2554.

[11] M. Shub. Global Stability of Dynamical Systems , 1986 .