Strange attractors are classified by bounding Tori

There is at present a doubly discrete classification for strange attractors of low dimension, d(L)<3. A branched manifold describes the stretching and squeezing processes that generate the strange attractor, and a basis set of orbits describes the complete set of unstable periodic orbits in the attractor. To this we add a third discrete classification level. Strange attractors are organized by the boundary of an open set surrounding their branched manifold. The boundary is a torus with g holes that is dressed by a surface flow with 2(g-1) singular points. All known strange attractors in R3 are classified by genus, g, and flow type.