Strange Attractors are Classified by Bounding Tori

Abstract

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 $R^3$ are classified by genus, $g$, and flow type.