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Topological entropy and partially hyperbolic diffeomorphisms. (English) Zbl 1143.37023
The authors consider continuity properties for the topological entropy of partially hyperbolic diffeomorphisms on compact manifolds. Fur this purpose, they define the notion of unstable and stable foliations stably carrying some nontrivial homologies. Under this topological assumption, the following two results are shown:
(1)
If the center foliation is one-dimensional, then the topological entropy is locally a constant.
(2)
If the center foliation is two-dimensional, then the topological entropy is continuous on the set of \(C^\infty\)-diffeomorphisms.
The proof uses a topological invariant, Yomdin’s theorem on upper semi-continuity, Katok’s theorem on lower semi-continuity for two-dimensional systems, as well as a refined Pesin-Ruelle inequality, which is shown for partially hyperbolic diffeomorphisms. In the final section, several examples illustrate that the above results do not hold without an appropriate assumption on the homology.

MSC:
37D30 Partially hyperbolic systems and dominated splittings
37B40 Topological entropy
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References:
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