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Topological Entropy and Partially Hyperbolic Diffeomorphisms

Yongxia Hua·Radu Saghin·Zhihong Xia
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Abstract

We consider partially hyperbolic diffeomorphisms on compact manifolds where the unstable and stable foliations stably carry some unique non-trivial homologies. 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 $C^\8$ 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.

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Yongxia Hua, Radu Saghin, Zhihong Xia. 2006-08-29. Topological Entropy and Partially Hyperbolic Diffeomorphisms. https://arxiv.org/abs/math/0608720

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