On the intersection of homoclinic classes on singular-hyperbolic sets

Abstract

We know that two different homoclinic classes contained in the samehyperbolic set are disjoint [12]. Moreover, a connectedsingular-hyperbolic attracting set with dense periodic orbits and aunique equilibrium is either transitive or the union of twodifferent homoclinic classes [6]. These results motivate thequestions of if two different homoclinic classes contained in thesame singular-hyperbolic set are disjoint or if the secondalternative in [6] cannot occur. Here we give a negativeanswer for both questions. Indeed we prove that every compact$3$-manifold supports a vector field exhibiting a connectedsingular-hyperbolic attracting set which has dense periodic orbits,a unique singularity, is the union of two homoclinic classes but isnot transitive.