TY - JOUR

T1 - Topological entropy and partially hyperbolic diffeomorphisms

AU - Hua, Yongxia

AU - Saghin, Radu

AU - Xia, Zhihong

PY - 2008/6

Y1 - 2008/6

N2 - 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 C∞ 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.

AB - 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 C∞ 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.

UR - http://www.scopus.com/inward/record.url?scp=44249124814&partnerID=8YFLogxK

U2 - 10.1017/S0143385707000405

DO - 10.1017/S0143385707000405

M3 - Article

AN - SCOPUS:44249124814

VL - 28

SP - 843

EP - 862

JO - Ergodic Theory and Dynamical Systems

JF - Ergodic Theory and Dynamical Systems

SN - 0143-3857

IS - 3

ER -