Shadowing Property of Non-Invertible Maps with Hyperbolic Measures

We show that if a differentiable map of a smooth manifold has a non-atomic ergodic hyperbolic measure then the topological entropy is positive and the space contains a hyperbolic horseshoe. we give some relations between hyperbolic measures and periodic points for differentiable maps. are generalized contents of the results obtained by Katok for diffeomorphisms.