  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
\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 $supp(\mu)$ of the given hyperbolic measure. Another is to show
Livshitz Theorem.
