On the intersection of sectional-hyperbolic sets

S. Bautista; C. A. Morales

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arXiv · 1410.0657

On the intersection of sectional-hyperbolic sets

S. Bautista·C. A. Morales

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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}).

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S. Bautista, C. A. Morales. 2014-10-02. On the intersection of sectional-hyperbolic sets. https://arxiv.org/abs/1410.0657

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