An extension of the Artin-Mazur theorem

1999/09/01 by Vadim Yu. Kaloshin · 1 citation
Mathematics · #math.DS

paper · pdf

published as Ann. of Math. (2) 150 (1999), no. 2, 729-741 · 13 pages, published version, abstract added in migration

arxiv created 1999/09/01 · arxiv updated 2009/11/30

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 fⁿ 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 dfⁿ 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 < ∞. Then the set of Artin-Mazur diffeomorphisms with only hyperbolic periodic orbits is dense in the space Diff^r(M).

Cited by

• On Kolmogorov-typical properties of symplectic dynamics

Related

• Semi periodic maps on complex manifolds
• A stretched exponential bound on the rate of growth of the number of periodic points for prevalent diffeomorphisms
• Generic family displaying robustly a fast growth of the number of periodic points
• Existence of periodic points with real and simple spectrum for diffeomorphisms in any dimension
• On the periodic structure of C¹ self-maps on the product of spheres of different dimensions