Skip to main content
×
Home
    • Aa
    • Aa

Contributions to the geometric and ergodic theory of conservative flows

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

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.

Copyright
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[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.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 19 *
Loading metrics...

Abstract views

Total abstract views: 77 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 20th September 2017. This data will be updated every 24 hours.