Hyperbolic Periodic Points and Hyperbolic Measures with Dominated Splitting

In this paper we consider a non-atomic invariant hyperbolic measure $μ$ of a $C^1$ diffeomorphsim on a compact manifold, in whose Oseledec splitting the stable bundle dominates the unstable bundle on $μ$ a.e. points. We show an \textit{exponentially} shadowing and an \textit{exponentially} closing lemma, and as applications we show two classical results. One is that there exists a hyperbolic periodic point such that the closure of its unstable manifold has \textit{positive} measure and it has a homoclinic point from which one can deduce a horseshoe. Moreover, such hyperbolic periodic points are dense in the support $supp(μ)$ of the given hyperbolic measure. Another is to show Livshitz Theorem.

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