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Zero-entropy invariant measures for skew product diffeomorphisms

Published online by Cambridge University Press:  17 July 2009

PENG SUN
PENG SUN*
Affiliation:
Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA (email: [email protected])
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Abstract

In this paper, we study some skew product diffeomorphisms with non-uniformly hyperbolic structure along fibers and show that there is an invariant measure with zero entropy which has atomic conditional measures along fibers. For such diffeomorphisms, our result gives an affirmative answer to the question posed by Herman as to whether a smooth diffeomorphism of positive topological entropy would fail to be uniquely ergodic. The proof is based on some techniques that are analogous to those developed by Pesin and Katok, together with an investigation of certain combinatorial properties of the projected return map on the base.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 30 , Issue 3 , June 2010 , pp. 923 - 930
Copyright
Copyright © Cambridge University Press 2009

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