On the hyperbolicity of flows

Abstract

Let X be a C1 vector field on a compact boundaryless Riemannian manifold M (dim M ≥ 2), and Λ a compact invariant set of X. Suppose that Λ has a hyperbolic splitting, i.e., TΛM = Es ⊕〈X〉⊕Eu with Es uniformly contracting and Eu uniformly expanding. We prove that if, in addition, Λ is chain transitive, then the hyperbolic splitting is continuous, i.e., Λ is a hyperbolic set. In general, when Λ is not necessarily chain transitive, the chain recurrent part is a hyperbolic set. Furthermore, we show that if the whole manifold M admits a hyperbolic splitting, then X has no singularity, and the flow is Anosov.