An extension of the Artin-Mazur theorem

الأصل · EN

Let M be a compact manifold. We call a mapping f in Cʳ(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 Cʳ(M,M) of mappings of M into itself, but only its open subset Diffʳ(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 Diffʳ(M).