Generic Diffeomorphisms with Superexponential Growth of Number of Periodic Orbits

Consider a compact manifold M of dimension at least 2 and the space of C^r-smooth diffeomorphisms Diff^r(M). The classical Artin-Mazur theorem says that for a dense subset D of Diff^r(M) the number of isolated periodic points grows at most exponentially fast (call it the A-M property). We extend this result and prove that diffeomorphisms having only hyperbolic periodic points with the A-M property are dense in Diff^r(M). Our proof of this result is much simpler than the original proof of Artin-Mazur. The second main result is that the A-M property is not (Baire) generic. Moreover, in a Newhouse domain ${\cal N} \subset Diff^r(M)$, an arbitrary quick growth of the number of periodic points holds on a residual set. This result follows from a theorem of Gonchenko-Shilnikov-Turaev, a detailed proof of which is also presented.