On axiom A diffeomorphisms C0-close to pseudo-Anosov maps

Axiom A diffeomorphisms g, C0-close to pseudo-Anosov maps f on a surface M and on its universal covering M are considered. It is shown that for “large” basic sets B of g, a necessary and sufficient condition to have the property that each trajectory in B is shadowed by a trajectory of f is to have a lifting to M of one of its stable (unstable) manifolds at a bounded distance from some stable (unstable) manifold of the lifting of f.