Super exponential divergence of periodic points for C^1-generic partially hyperbolic homoclinic classes

A diffeomorphism f is called super exponential divergent if for every r>1, the lower limit of #Per_n(f)/r^n diverges to infinity as n tends to infinity, where Per_n(f) is the set of all periodic points of f with period n. This property is stronger than the usual super exponential growth of the number of periodic points. We show that for a three dimensional manifold M, there exists an open subset O of Diff^1(M) such that diffeomorphisms with super exponential divergent property form a dense subset of O in the C^1-topology. A relevant result of non super exponential divergence for diffeomorphisms in a locally generic subset of Diff^r(M) (r=1,2,...\infty) is also shown.