On the intersection of sectional-hyperbolic sets

Serafin  Bautista; Carlos A.  Morales

doi:10.3934/jmd.2015.9.203

Article Contents

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

This issue Previous Article Local rigidity of homogeneous actions of parabolic subgroups of rank-one Lie groups Next Article Hofer's length spectrum of symplectic surfaces

On the intersection of sectional-hyperbolic sets

Serafin Bautista^1, and
Carlos A. Morales^2,

1.
Universidad Nacional de Colombia, Depto. de Matemáticas, Facultad de Ciencias, Bogota
2.
Instituto de Matemática, Universidade Federal do Rio de Janeiro,, P. O. Box 68530, 21945-970 Rio de Janeiro

Received: October 2014

Revised: June 2015

Published: September 2015

Abstract / Introduction Related Papers Cited by

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]).

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Mathematics Subject Classification: Primary: 37D30; Secondary: 37C10.

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