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Topological entropy and partially hyperbolic diffeomorphisms

Published online by Cambridge University Press:  01 June 2008

YONGXIA HUA ,
RADU SAGHIN  and
ZHIHONG XIA
YONGXIA HUA
Affiliation:
Department of Mathematics, Northwestern University, Evanston, Illinois 60208, USA (email: hua@math.northwestern.edu, xia@math.northwestern.edu)
RADU SAGHIN
Affiliation:
Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 2E4 (email: rsaghin@fields.utoronto.ca)
ZHIHONG XIA
Affiliation:
Department of Mathematics, Northwestern University, Evanston, Illinois 60208, USA (email: hua@math.northwestern.edu, xia@math.northwestern.edu)
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Abstract

We consider partially hyperbolic diffeomorphisms on compact manifolds. We define the notion of the unstable and stable foliations stably carrying some unique non-trivial homologies. Under this topological assumption, we prove the following two results: if the center foliation is one-dimensional, then the topological entropy is locally a constant; and if the center foliation is two-dimensional, then the topological entropy is continuous on the set of all diffeomorphisms. The proof uses a topological invariant we introduced, Yomdin’s theorem on upper semi-continuity, Katok’s theorem on lower semi-continuity for two-dimensional systems, and a refined Pesin–Ruelle inequality we proved for partially hyperbolic diffeomorphisms.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 28 , Issue 3 , June 2008 , pp. 843 - 862
Copyright
Copyright © Cambridge University Press 2008

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