Hostname: page-component-76fb5796d-25wd4 Total loading time: 0 Render date: 2024-04-27T00:32:29.668Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  06 September 2012

JIANYU CHEN ,
HUYI HU  and
YAKOV PESIN
JIANYU CHEN
Affiliation:
Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA
HUYI HU
Affiliation:
Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA
YAKOV PESIN
Affiliation:
Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

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.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 33 , Issue 6 , December 2013 , pp. 1748 - 1785
Copyright
Copyright © 2012 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Burns, K., Dolgopyat, D. and Pesin, Ya.. Partial hyperbolicity, Lyapunov exponents and stable ergodicity. J. Stat. Phys. 108(5–6) (2002), 927–942.Google Scholar
[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.Google Scholar
[3]Bolsinov, A. and Taimanov, I.. Integrable geodesic flows with positive topological entropy. Invent. Math. 140(3) (2000), 639–650.Google Scholar
[4]Burns, K. and Wilkinson, A.. On the ergodicity of partially hyperbolic systems. Ann. of Math. (2) 171 (2010), 451–489.Google Scholar
[5]Chen, J., Hu, H. and Pesin, Ya.. The essential coexistence phenomenon in dynamics. Dyn. Syst. (2013) to appear.CrossRefGoogle Scholar
[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.Google Scholar
[7]Dolgopyat, D. and Pesin, Ya.. Every compact manifold carries a completely hyperbolic diffeomorphism. Ergod. Th. & Dynam. Sys. 22 (2002), 409–435.Google Scholar
[8]Donnay, V.. Geodesic flow on the two-sphere. I. Positive measure entropy. Ergod. Th. & Dynam. Sys. 8(4) (1988), 531–553.CrossRefGoogle Scholar
[9]Hirsch, M., Pugh, C. and Shub, M.. Invariant Manifolds (Lecture Notes in Mathematics, 583). Springer, Berlin, 1977.Google Scholar
[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.Google Scholar
[11]Kornfeld, I., Sinai, Ya. and Fomin, S.. Ergodic Theory. Springer, New York, 1982.CrossRefGoogle Scholar
[12]Palis, J. and de Melo, W.. Geometric Theory of Dynamical Systems. Springer, New York, 1982.CrossRefGoogle Scholar
[13]Pesin, Ya.. Lectures on Partial Hyperbolicity and Stable Ergodicity (Zürich Lectures in Advanced Mathematics). European Mathematical Society, Zürich, 2004.Google Scholar
[14]Pugh, C. and Shub, M.. Stably ergodic dynamical systems and partial hyperbolicity. J. Complexity 13 (1997), 125–179.CrossRefGoogle Scholar
[15]Shub, M. and Wilkinson, A.. Pathological foliations and removable zero exponents. Invent. Math. 139(3) (2000), 495–508.Google Scholar