  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).
