Axiom A flows with a transverse torus

Abstract

Let X X be an Axiom A flow with a transverse torus T T exhibiting a unique orbit O O that does not intersect T T . Suppose that there is no null-homotopic closed curve in T T contained in either the stable or unstable set of O O . Then we show that X X has either an attracting periodic orbit or a repelling periodic orbit or is transitive. In particular, an Anosov flow with a transverse torus is transitive if it has a unique periodic orbit that does not intersect the torus.