Topological Entropy and Partially Hyperbolic Diffeomorphisms

2006/08/29 by Hua, Yongxia, Saghin, Radu, Xia, Zhihong · 2 citations
#37A35 #37C40 #37D25 #37D30 #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.math/0608720

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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