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

On the intersection of homoclinic classes on singular-hyperbolic sets

S. Bautista 1, , C. Morales 2, and  M. J. Pacifico 2,

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.
Keywords: Homoclinic Class, Singular-hyperbolic Set., Attracting Set.
Mathematics Subject Classification: Primary: 37D30, Secondary: 37D4.
Citation: S. Bautista, C. Morales, M. J. Pacifico. On the intersection of homoclinic classes on singular-hyperbolic sets. Discrete and Continuous Dynamical Systems, 2007, 19 (4) : 761-775. doi: 10.3934/dcds.2007.19.761
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