Preprint D15/2005
Intersection of homoclinic classes on singular-hyperbolic sets

Maria Jose Pacifico | Bautista, Serafin | Morales, Carlos Arnoldo

**Keywords: **
Homoclinic Class | Singular-hyperbolic Set | Vector Field

We know that two different homoclinic classes
contained in the same hyperbolic set are disjoint \cite{N}.
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
\cite{MPaa}.
These results motivate the questions if
two different homoclinic classes contained in the same singular-hyperbolic set
are disjoint or if the second alternative in \cite{MPaa}
cannot occur.
Here we give 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.