Generic diffeomorphisms with superexponential growth of number of periodic orbits

Let M be a smooth compact manifold of dimension at least 2 and Diff(r)(M) be the space of C-r smooth diffeomorphisms of M. Associate to each diffeomorphism f is an element of Diff(r) (M) the sequence P-n(f) of the number of isolated periodic points for f of period n. In this paper we exhibit an open set N in the space of diffeomorphisms Diff(r)' (M) such for a Baire generic diffeomorphism f is an element of N the number of periodic points P(n)f grows with a period n faster than any following sequence of numbers {a(n)}(n is an element of Z+) along a subsequence, i.e. P-n(f) > a(ni) for some n(i) --> infinity with i --> infinity. In the cases of surface diffeomorphisms, i.e. dim M = 2, an open set N with a supergrowth of the number of periodic points is a Newhouse domain. A proof of the man result is based on the Gontchenko-Shilnikov-Turaev Theorem [GST]. A complete proof of that theorem is also presented.