Singular-Hyperbolic Attractors with Handlebody Basins

Abstract

An attractor of a vector field X with generating flow X t is a transitive set equal to $${\bigcap\limits_{t > 0} {X_{t} {\left( U \right)}} }$$ for some neighborhood U called basin of attraction. An attractor is singular-hyperbolic if it has singularities (all hyperbolic) and is partially hyperbolic with volume expanding direction. A singular-hyperbolic repeller is a singular-hyperbolic attractor for the time reversed flow. A handlebody of genus $$n \in \mathbb{N}$$ 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. We show that every orientable handlebody of genus n???2 can be realized as the basin of attraction of a singular-hyperbolic attractor. 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 every closed 3-manifold, r???1.