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Contributions to the geometric and ergodic theory of conservative flows

  • MÁRIO BESSA (a1) and JORGE ROCHA (a2)

We prove the following dichotomy for vector fields in a $C^1$-residual subset of volume-preserving flows: for Lebesgue-almost every point, either all of its Lyapunov exponents are equal to zero or its orbit has a dominated splitting. Moreover, we prove that a volume-preserving and $C^1$-stably ergodic flow can be $C^1$-approximated by another volume-preserving flow which is non-uniformly hyperbolic.

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[2]V. Araújo and M. Bessa. Dominated splitting and zero volume for incompressible three-flows. Nonlinearity 21 (2008), 16371653.

[3]V. Araújo and M. J. Pacifico. Three-Dimensional Flows (Ergebnisse der Mathematik und ihrer Grenzgebiete, 53). Springer, Berlin, 2010.

[5]L. Arnold. Random Dynamical Systems. Springer, Berlin, 1998.

[6]A. Avila. On the regularization of conservative maps. Acta Math. 205 (2010), 518.

[8]M. Bessa. Dynamics of generic multidimensional linear differential systems. Adv. Nonlinear Stud. 8 (2008), 191211.

[9]M. Bessa and J. L. Dias. Generic dynamics of $4$-dimensional $C^2$ Hamiltonian systems. Comm. Math. Phys. 281 (2008), 597619.

[10]M. Bessa and P. Duarte. Abundance of elliptic dynamics on conservative 3-flows. Dyn. Syst. 23(4) (2008), 409424.

[11]M. Bessa and J. Rocha. Removing zero Lyapunov exponents in volume-preserving flows. Nonlinearity 20 (2007), 10071016.

[12]M. Bessa and J. Rocha. On $C^1$-robust transitivity of volume-preserving flows. J. Differential Equations 245(11) (2008), 31273143.

[16]J. Bochi, B. Fayad and E. Pujals. A remark on conservative diffeomorphisms. C. R. Math. Acad. Sci. Paris Ser. I 342 (2006), 763766.

[17]J. Bochi and M. Viana. The Lyapunov exponents of generic volume-preserving and symplectic maps. Ann. of Math. (2) 161(3) (2005), 14231485.

[22]C. Ferreira. Stability properties of divergence-free vector fields. Dyn. Syst. 27(2) (2012), 223238.

[23]J. Franks. Necessary conditions for the stability of diffeomorphisms. Trans. Amer. Math. Soc. 158 (1971), 301308.

[24]S. Gan and L. Wen. Nonsingular star flows satisfy Axiom A and the no-cycle condition. Invent. Math. 164(2) (2006), 279315.

[26]R. Johnson, K. Palmer and G. Sell. Ergodic properties of linear dynamical systems. SIAM J. Math. Anal. 18 (1987), 133.

[30]J. Moser. On the volume elements on a manifold. Trans. Amer. Math. Soc. 120 (1965), 286294.

[33]C. Pugh and M. Shub. Stable ergodicity. Bull. Amer. Math. Soc. (N.S.) 41(1) (2004), 141 (with an appendix by Alexander Starkov).

[34]C. Robinson. Generic properties of conservative systems. Amer. J. Math. 92 (1970), 562603.

[36]M. Viana. Almost all cocycles over any hyperbolic system have non-vanishing Lyapunov exponents. Ann. of Math. (2) 167 (2008), 643680.

[37]C. Zuppa. Regularisation $C^{\infty }$ des champs vectoriels qui préservent l’elément de volume. Bol. Soc. Bras. Mat. 10(2) (1979), 5156.

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Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
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