Preprint A254/2003
Singular-hyperbolic attractors with handlebody basins

C. A. Morales

**Keywords: **
Handlebody | Partially Hyperbolic Set | Heegaard Splitting

A {\em handlebody of genus $n\in I\!\! N$}
(or cube with $n$-handles)
is a compact $3$-manifold $V$ containing a disjoint
collection of $n$ properly embedded $2$-cells such that the result of cutting $V$ along these disks is a $3$-cell (\cite{He} p. 15).
An {\em attractor} of a vector field $X$
with generating flow $X_t$ is a
transitive set of it equals to $\cap_{t>0}X_t(U)$
for some neighborhood $U$ called {\em isolating block}.
A partially hyperbolic attractor is
{\em singular-hyperbolic} if it contains singularities
(all hyperbolic) and has volume expanding central direction \cite{MPP}. A {\em singular-hyperbolic
repeller} is a singular-hyperbolic attractor for the time-reversed vector field.
We show that every orientable handlebody of genus
$n\geq 2$ can be realized as the isolating block of a singular-hyperbolic attractor with $n-1$ singularities.
Hence every closed orientable $3$-manifold supports
a vector field whose nonwandering set
consists of a singular-hyperbolic attractor
and a singular-hyperbolic repeller.
In particular,
there are open sets of $C^r$ vector fields
without hyperbolic attractors or hyperbolic repellers
on {\em every}
closed $3$-manifold,
$r\geq 1$.