An extension of the Artin-Mazur theorem

Vadim Yu. Kaloshin

Review from Zentralblatt MATH:

Given a compact manifold $M$, in 1965 Artin and Mazur showed that there is a dense set of endomorphisms on $M$ for which the number of isolated periodic points grows at most exponentially with the period. Call a map with this property an Artin-Mazur map. The author extends the result by showing that the set of Artin-Mazur diffeomorphisms on $M$ with only hyperbolic periodic points lies dense in the set of diffeomorphisms on $M$. In the arguments, the manifold $M$ is embedded into Euclidean space and a map $F$ on a tubular neighborhood of $M$ that extends $f$ is approximated by a polynomial map, after which periodic points of polynomial maps are studied. By considering diffeomorphisms with persistent homoclinic tangencies, it is further shown that the set of Artin-Mazur diffeomorphisms is not residual.

Reviewed by Ale Jan Homburg

Keywords: Artin-Mazur map; Artin-Mazur diffeomorphisms; hyperbolic periodic points; persistent homoclinic tangencies

Classification (MSC2000): 37C20 37G20

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