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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
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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 

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