Institute for Mathematical Sciences

Preprint ims99-2

V. Kaloshin
Generic Diffeomorphisms with Superexponential Growth of Number of Periodic Orbits

Abstract: 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 ${\cal N} \subset \textup{Diff}^r(M)$, 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.

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