Journal of Modern Dynamics (JMD)

On the intersection of sectional-hyperbolic sets

Pages: 203 - 218, Volume 9, 2015      doi:10.3934/jmd.2015.9.203

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Serafin Bautista - Universidad Nacional de Colombia, Depto. de Matemáticas, Facultad de Ciencias, Bogota, Colombia (email)
Carlos A. Morales - Instituto de Matemática, Universidade Federal do Rio de Janeiro,, P. O. Box 68530, 21945-970 Rio de Janeiro, Brazil (email)

Abstract: 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]).

Keywords:  Sectional-hyperbolic set, star flow, singularity
Mathematics Subject Classification:  Primary: 37D30; Secondary: 37C10.

Received: October 2014;      Revised: June 2015;      Available Online: September 2015.