摘要
Let ■ be the linear space of all C<sup>1</sup> vector fields X on a compact n-dimensionalC<sup>∞</sup> Riemann manifold（n≥2）,endowed with the C<sup>1</sup> norm ‖X‖<sub>1</sub>.Write θ（X）for the numberof contractible periodic orbits of X∈（?）,which may be finite or infinite.Let （?）<sup>*</sup> 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∈（?）.
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∈(?).
基金
Supported by National Natural Science Foundation of China
