Classifying the expanding attractors on the figure‐eight knot exterior and the non‐transitive Anosov flows on the Franks–Williams manifold

Abstract

The figure‐eight knot exterior N0$N_0$ supports a natural DA (derived from Anosov) expanding attractor, with which Franks–Williams constructed the first example of non‐transitive Anosov flow. This flow lies in a 3‐manifold M0$M_0$ which is the double of N0$N_0$ . We call M0$M_0$ by the Franks–Williams manifold. In this paper, we prove that, up to orbit‐equivalence, this DA expanding attractor is the unique expanding attractor supported by N0$N_0$ . We also show that, up to orbit‐equivalence, the non‐transitive Anosov flow constructed by Franks and Williams is the unique non‐transitive Anosov flow supported by M0$M_0$ . We also extend these results to a more general context.