Strange attractors with topologically simple basins

We consider the basin WsΛ of a 1-dimensional hyperbolic attractor Λ in an m-dimensional manifold M where m⩾4, WsΛ is orientable and Λ is transversely tame. is shown that there is a proper open subset G of the 3-dimensional sphere S3 such that WsΛ is homeomorphic to G×Rm−3. me conditions concerning the intrinsic structure of Λ are given each of which is necessary and sufficient for WsΛ being an open handlebody. other condition concerning the intrinsic structure of Λ is given which implies that the set G mentioned above can be chosen to be S3 minus the union of a finite number of simple closed curves with a single common point z which are infinitely linked or knotted at z. m=3 or if Λ is not transversely tame corresponding assertions hold, if WsΛ is replaced by WsΛ×R.