ON(η,d)-CONTRACTIBLE ORBITS OF VECTOR FIELDS

Let ■ be the linear space of all C~1 vector fields X on a compact n-dimensionalC~∞ Riemann manifold(n≥2),endowed with the C~1 norm ‖X‖_1.Write θ(X)for the numberof contractible periodic orbits of X∈(?),which may be finite or infinite.Let (?)~* be the set ofall X∈(?) possessing the property that X has a neighbourhood (?) such that every Y∈(?) hasonly a finite number of singularities and at most a countable number of periodic orbits.Inthis paper,it is shown that any given S∈(?) has a neighbourhood (?) in (?) together with anumber λ=λ(?)>0 such that θ(X)≤λfor all X∈(?).