We consider a compact locally maximal invariant set K of a C²-diffeomorphism f:M→M on a smooth Riemannian manifold M. We study the topological pressure (with respect to f|K) for a wide class of Hölder continuous potentials. Under a mild nonuniform hyperbolicity assumption we show that the pressure of an appropriate potential is entirely determined by its values on periodic orbits which have sufficiently strong hyperbolic behavior.
We also consider the case of the potential that relates to the volume growth rate and study its saddle point pressure.