Shadowing by Non Uniformly Hyperbolic Periodic Points and Uniform Hyperbolicity

We prove that, under a mild condition on the hyperbolicity of its periodic points, a map g which is topologically conjugated to a hyperbolic map (respectively, an expanding map) is also a hyperbolic map (respectively, an expanding map). In particular, this result gives a partial positive answer for a question done by A. Katok, in a related context.