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On the intersection of sectional-hyperbolic sets

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  • We study the intersection of a positively sectional-hyperbolic set and a negatively sectional-hyperbolic set of a flow on a compact manifold. Indeed, we show that such an intersection is not a hyperbolic set in general. Next, we show that such an intersection is a hyperbolic set if the sets involved in the intersection are both transitive. In general, we prove that such an intersection is the disjoint union of a nonsingular hyperbolic set, a finite set of singularities, and a set of regular orbits joining these dynamical objects. Finally, we exhibit a three-dimensional star flow with a positively (but not negatively) sectional-hyperbolic homoclinic class and a negatively (but not positively) sectional-hyperbolic homoclinic class whose intersection is a periodic orbit. This provides a counterexample to a conjecture of Shi, Zhu, Gan and Wen ([25], [26]).
    Mathematics Subject Classification: Primary: 37D30; Secondary: 37C10.


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