On the intersection of homoclinic classes on singular-hyperbolic sets

S. Bautista; C. Morales; M. J. Pacifico

doi:10.3934/dcds.2007.19.761

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2007, Volume 19, Issue 4: 761-775. Doi: 10.3934/dcds.2007.19.761

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On the intersection of homoclinic classes on singular-hyperbolic sets

1.
Departamento de Matemticas, Universidad Nacional de Colombia, Bogot, D.C.
2.
Instituto de Matemática, Universidade Federal do Rio de Janeiro, P. O. Box 68530, 21945-970, Rio de Janeiro

Received: December 31, 2005

Revised: May 31, 2007

Published: September 2007

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Abstract

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.

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Mathematics Subject Classification: Primary: 37D30, Secondary: 37D45.

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