Consider a compact manifold M of dimension at least 2 and the space of $C^r$-smooth diffeomorphisms $\mathrm{Diff}^r(M)$. The classical Artin-Mazur theorem says that for a dense subset D of $\mathrm{Diff}^r(M)$ 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 $\mathrm{Diff}^r(M)$. 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 $\mathcal{N} \subset \mathrm{Diff}^r(M)$, 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.