Classifying expanding attractors on figure eight knot complement space and non-transitive Anosov flows on Franks-Williams manifold

TL;DR

This paper classifies unique expanding attractors on the figure eight knot complement space and non-transitive Anosov flows on the Franks-Williams manifold, establishing their uniqueness up to orbit-equivalence and exploring more general cases.

Contribution

It proves the uniqueness of the DA expanding attractor on $N_0$ and the non-transitive Anosov flow on $M_0$, extending to a family of related manifolds.

Findings

DA attractor is unique on $N_0$

Non-transitive Anosov flow is unique on $M_0$

Complete classification for a family of toroidal 3-manifolds

Abstract

The path closure of figure eight knot complement space, $N_0$, supports a natural DA (derived from Anosov) expanding attractor. Using this attractor, Franks-Williams constructed the first example of non-transitive Anosov flow on the manifold $M_0$ obtained by gluing two copies of $N_0$ through identity map along their boundaries, named by Franks-Williams manifold. In this paper, our main goal is to classify expanding attractors on $N_0$ and non-transitive Anosov flows on $M_0$. We prove that, up to orbit-equivalence, the DA expanding attractor is the unique expanding attractor supported by $N_0$, and the non-transitive Anosov flow constructed by Franks and Williams is the unique non-transitive Anosov flow admitted by $M_0$. Moreover, more general cases are also discussed. In particular, we completely classify non-transitive Anosov flows on a family of infinitely many toroidal $3$-manifolds with two hyperbolic pieces, obtained by gluing two copies of $N_0$ through any gluing homeomorphism.