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An extension of the Artin-Mazur theorem

Vadim Yu. Kaloshin
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Abstract

Let M be a compact manifold. We call a mapping f in C^r(M,M) an Artin-Mazur mapping if the number of isolated periodic points of f^n grows at most exponentially in n. Artin and Mazur posed the following problem: What can be said about the set of Artin-Mazur mappings with only transversal periodic orbits? Recall that a periodic orbit of period n is called transversal if the linearization df^n at this point has for an eigenvalue no nth roots of unity. Notice that a hyperbolic periodic point is always transversal, but not vice versa. We consider not the whole space C^r(M,M) of mappings of M into itself, but only its open subset Diff^r(M). The main result of this paper is the following theorem: Let 1 <= r < \infty. Then the set of Artin-Mazur diffeomorphisms with only hyperbolic periodic orbits is dense in the space Diff^r(M).

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Vadim Yu. Kaloshin. 1999-09-01. An extension of the Artin-Mazur theorem. https://arxiv.org/abs/math/9909196

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