This dissertation addresses the continuity property of topological entropy for partially hyperbolic diffeomorphisms with one-dimension center. In [1], Saghin, Xia and I proved this with the additional topological assumption of unique non-zero homologies on strong stable and strong unstable foliations. Actually, in [1], we proved the entropy is locally constant for this class of diffeomorphisms. The purpose of this paper is to prove the continuity of entropy at time-one maps of transitive Anosov flows, a class of partially hyperbolic diffeomorphisms with one-dimension center, which does not satisfy the additional topological assumption.
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