Generic Diffeomorphisms with Superexponential Growth of Number of Periodic Orbits

Let M be a smooth compact manifold of dimension at least 2 and Diffr(M) be the space of Cr smooth diffeomorphisms of M. Associate to each diffeomorphism f;isin; Diffr(M) the sequence Pn(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 Diffr(M) such for a Baire generic diffeomorphism f∈N the number of periodic points Pnf grows with a period n faster than any following sequence of numbers {an}n∈Z+ along a subsequence, i.e. Pn(f)>ani for some ni→∞ with i→∞. 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.