2014/10/02 by S. Bautista, Bautista, S., C. A. Morales +1
Mathematics · Physics and Astronomy · #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Primary: 37D20 #Quantum chaos and dynamical systems #Secondary: 37C70 #math.DS #msc:37C70 #msc:37D20
paper · pdf · doi:10.48550/arxiv.1410.0657
15 pages, 9 figures. Results announced in the {\em I Workshop on Sectional-Anosov flows} which took place in September 22 of 2014 at the Federal University of Viçosa-MG, Brasil
arxiv created 2014/10/02 · openalex publication_date 2014/10/02 · arxiv updated 2014/10/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 (\citesgw, \citezgw).