On the intersection of sectional-hyperbolic sets

Abstract

We analyse the intersection of positively and negatively sectional-hyperbolic sets for flows on compact manifolds. First we prove that such an intersection is hyperbolic if the intersecting sets are both transitive (this is false without such a hypothesis). Next we prove that, in general, such an intersection consists of a nonsingular hyperbolic set, finitely many singularities and regular orbits joining them. Afterward we exhibit a three-dimensional star flow with two homoclinic classes, one being positively (but not negatively) sectional-hyperbolic and the other negatively (but not positively) sectional-hyperbolic, whose intersection reduces to a single periodic orbit. This provides a counterexample to a conjecture by Shy, Zhu, Gan and Wen (\cite{sgw}, \cite{zgw}).