December  2007, 19(4): 761-775. doi: 10.3934/dcds.2007.19.761

On the intersection of homoclinic classes on singular-hyperbolic sets

1. 

Departamento de Matemticas, Universidad Nacional de Colombia, Bogot, D.C., Colombia

2. 

Instituto de Matemática, Universidade Federal do Rio de Janeiro, P. O. Box 68530, 21945-970, Rio de Janeiro, Brazil, Brazil

Received  January 2006 Revised  June 2007 Published  September 2007

We know that two different homoclinic classes contained in the same hyperbolic set are disjoint [12]. Moreover, a connected singular-hyperbolic attracting set with dense periodic orbits and a unique equilibrium is either transitive or the union of two different homoclinic classes [6]. These results motivate the questions of if two different homoclinic classes contained in the same singular-hyperbolic set are disjoint or if the second alternative in [6] cannot occur. Here we give a negative answer for both questions. Indeed we prove that every compact $3$-manifold supports a vector field exhibiting a connected singular-hyperbolic attracting set which has dense periodic orbits, a unique singularity, is the union of two homoclinic classes but is not transitive.
Citation: S. Bautista, C. Morales, M. J. Pacifico. On the intersection of homoclinic classes on singular-hyperbolic sets. Discrete & Continuous Dynamical Systems, 2007, 19 (4) : 761-775. doi: 10.3934/dcds.2007.19.761
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