Generic Diffeomorphisms with Superexponential Growth of Number of Periodic Orbits

Abstract:

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;isin; 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∈N the number of periodic points P _n f grows with a period n faster than any following sequence of numbers {a _n } _{n ∈ Z +} along a subsequence, i.e. P _n (f)>a _ni for some n _i →∞ 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.