A property of vector fields without singularity in {\mathcal G}^1(M)

Abstract

We prove the following property: if X in \mathcal G^1(M) has no singularity and x \in \Sigma(X), then \overline{\operatorname{orbit}(x)} \cap \overline{\operatorname{per}(X)} \not = \emptyset. In addition, if we assume \overline{\operatorname{per}_i(X)} \cap \overline{\operatorname{per}_j(X)} = \emptyset for i \not = j, then \overline{\operatorname{per}(X)} = \bigcup_{i=0}^{n-1} \overline{\operatorname{per}_i(X)} is a hyperbolic set. Moreover, we shall give a proof of the \Omega-stability conjecture for flows.