Hyperbolic Periodic Points and Hyperbolic Measures with Dominated Splitting

In this paper we consider a non-atomic invariant hyperbolic measure $\mu$ of a $C^1$ diffeomorphsim on a compact manifold, in whose Oseledec splitting the stable bundle dominates the unstable bundle on $\mu$ a.e. points. We show an $\mathit{\text{exponentially}}$ shadowing and an $\mathit{\text{exponentially}}$ 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 $\mathit{\text{positive}}$ measure and it has a homoclinic point from which one can deduce a horseshoe. Moreover, such hyperbolic periodic points are dense in the support $supp(\mu )$ of the given hyperbolic measure. Another is to show Livshitz Theorem.