Skip to main content Accessibility help

A volume preserving flow with essential coexistence of zero and non-zero Lyapunov exponents

  • JIANYU CHEN (a1), HUYI HU (a2) and YAKOV PESIN (a1)

We demonstrate essential coexistence of hyperbolic and non-hyperbolic behavior in the continuous-time case by constructing a smooth volume preserving flow on a five-dimensional compact smooth manifold that has non-zero Lyapunov exponents almost everywhere on an open and dense subset of positive but not full volume and is ergodic on this subset while having zero Lyapunov exponents on its complement. The latter is a union of three-dimensional invariant submanifolds, and on each of these submanifolds the flow is linear with Diophantine frequency vector.

Hide All
[1]Burns, K., Dolgopyat, D. and Pesin, Ya.. Partial hyperbolicity, Lyapunov exponents and stable ergodicity. J. Stat. Phys. 108(5–6) (2002), 927942.
[2]Barreira, L. and Pesin, Ya.. Nonuniform Hyperbolicity: Dynamics of Systems with Nonzero Lyapunov Exponents (Encyclopedia of Mathematics and its Applications, 115). Cambridge University Press, Cambridge, 2007.
[3]Bolsinov, A. and Taimanov, I.. Integrable geodesic flows with positive topological entropy. Invent. Math. 140(3) (2000), 639650.
[4]Burns, K. and Wilkinson, A.. On the ergodicity of partially hyperbolic systems. Ann. of Math. (2) 171 (2010), 451489.
[5]Chen, J., Hu, H. and Pesin, Ya.. The essential coexistence phenomenon in dynamics. Dyn. Syst. (2013) to appear.
[6]Dolgopyat, D., Hu, H. and Pesin, Ya.. An example of a smooth hyperbolic measure with countably many ergodic components. Smooth Ergodic Theory and Its Applications (Seattle, 1999) (Proceedings of Symposia in Pure Mathematics, 69). American Mathematical Society, Providence, RI, 2001.
[7]Dolgopyat, D. and Pesin, Ya.. Every compact manifold carries a completely hyperbolic diffeomorphism. Ergod. Th. & Dynam. Sys. 22 (2002), 409435.
[8]Donnay, V.. Geodesic flow on the two-sphere. I. Positive measure entropy. Ergod. Th. & Dynam. Sys. 8(4) (1988), 531553.
[9]Hirsch, M., Pugh, C. and Shub, M.. Invariant Manifolds (Lecture Notes in Mathematics, 583). Springer, Berlin, 1977.
[10]Hu, H., Pesin, Ya. and Talitskaya, A.. A volume preserving diffeomorphism with essential coexistence of zero and nonzero Lyapunov exponents. Comm. Math. Phys. (2012) to appear.
[11]Kornfeld, I., Sinai, Ya. and Fomin, S.. Ergodic Theory. Springer, New York, 1982.
[12]Palis, J. and de Melo, W.. Geometric Theory of Dynamical Systems. Springer, New York, 1982.
[13]Pesin, Ya.. Lectures on Partial Hyperbolicity and Stable Ergodicity (Zürich Lectures in Advanced Mathematics). European Mathematical Society, Zürich, 2004.
[14]Pugh, C. and Shub, M.. Stably ergodic dynamical systems and partial hyperbolicity. J. Complexity 13 (1997), 125179.
[15]Shub, M. and Wilkinson, A.. Pathological foliations and removable zero exponents. Invent. Math. 139(3) (2000), 495508.
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? *


Full text views

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

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed