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, $r\geq 1$.