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.

Anexos:

• shf4.ps