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Annals of Mathematics, II. Series Vol. 150, No. 2, pp. 729741 (1999) 

An extension of the ArtinMazur theoremVadim Yu. KaloshinReview 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 ArtinMazur map. The author extends the result by showing that the set of ArtinMazur 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 ArtinMazur diffeomorphisms is not residual. Reviewed by Ale Jan Homburg Keywords: ArtinMazur map; ArtinMazur diffeomorphisms; hyperbolic periodic points; persistent homoclinic tangencies Classification (MSC2000): 37C20 37G20 Full text of the article:
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