We study entropies caused by the unstable part of partially hyperbolic systems. We define unstable metric entropy and unstable topological entropy, and establish a variational principle for partially hyperbolic diffeomorphsims, which states that the unstable topological entropy is the supremum of the unstable metric entropy taken over all invariant measures. The unstable metric entropy for an invariant measure is defined as a conditional entropy along unstable manifolds, and it turns out to be the same as that given by Ledrappier-Young, though we do not use increasing partitions. The unstable topological entropy is defined equivalently via separated sets, spanning sets and open covers along a piece of unstable leaf, and it coincides with the unstable volume growth along unstable foliation. We also obtain some properties for the unstable metric entropy such as affineness, upper semi-continuity and a version of Shannon-McMillan-Breiman theorem.
Link to this article: http://journals.cambridge.org/abstract_S0143385707000405 How to cite this article: YONGXIA HUA, RADU SAGHIN and ZHIHONG XIA (2008). Topological entropy and partially hyperbolic diffeomorphisms.Abstract. We consider partially hyperbolic diffeomorphisms on compact manifolds. We define the notion of the unstable and stable foliations stably carrying some unique nontrivial 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.
We show that for any C 1 partially hyperbolic diffeomorphism, there is a full volume subset, such that any Cesaro limit of any point in this subset satisfies the Pesin formula for partial entropy.This result has several important applications. First we show that for any C 1+ partially hyperbolic diffeomorphism with one dimensional center, there is a full volume subset, such that every point in this set belongs to either the basin of a physical measure with non-vanishing center exponent, or the center exponent of any limit of the sequence 1 n n−1 i=0 δ f i (x) is vanishing. We also prove that for any diffeomorphism with mostly contracting center, it admits a C 1 neighborhood such that every diffeomorphism in a C 1 residual subset of this open set admits finitely many physical measure, whose basins have full volume.Remark 1.1. As a direct consequence, if G u (f ) consists of a unique measure, then this measure is automatically a physical measure whose basin has full volume.The partially hyperbolic diffeomorphisms with mostly contracting center, which were first studied in [5], are those C 1+ partially hyperbolic diffeomorphisms whose Gibbs u-states only have negative center Lyapunov exponents.Mostly contracting diffeomorphisms contain an abundance of systems. See for example [5] and [10]. It is also shown in [27] that mostly contracting is a C 1 open property. Moreover, if f is a C 1+ partially hyperbolic volume preserving diffeomorphisms with one dimensional center, such that the volume has negative center exponent and the unstable foliation is minimal, then f is mostly contracting.
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