Nonsingular structural stable chaotic 3-flows of attractor-repeller type

TL;DR

This work proves that every closed orientable 3-manifold $M$ carries a structurally stable nonsingular flow of attractor–repeller type, whose non-wandering set $NW(f^t)$ is a two-dimensional expanding attractor together with finitely many repelling periodic trajectories. It achieves this by starting from a canonical flow on $\mathbb{S}^3$ with $NW(f_0^t)=l_1\cup\Lambda_a$ (where $l_1$ is the figure-eight repeller and $\Lambda_a$ is a 2D expanding attractor) and lifting it to arbitrary $M$ via $m$-fold branched coverings branched over a link, yielding $NW(f^t)=p^{-1}(NW(f_0^t))$. The results further show that on $\mathbb{S}^3$ the repellers can realize any link containing the figure-eight knot, while a single repeller cannot be a torus knot nor be trivial, with these restrictions derived from MS-foliations, taut foliations, and Novikov theory. Altogether, the paper broadens the class of 3-manifolds admitting attractor–repeller flows and provides a systematic method to realize prescribed repeller links via branched coverings.

Abstract

We show that any orientable closed 3-manifold $M$ admits structurally stable non-singular flow $f^t$ whose non-wandering set $NW(f^t)$ consists of a 2-dimensional expanding attractor and finitely many repelling periodic trajectories. For $M=\mathbb{S}^3$, we prove that the set of repelling periodic trajectories can be an arbitrary link provided that this link contains the figure eight knot. When a link consists of a unique repelling periodic trajectory (not necessarily a figure eight knot), we prove that this trajectory cannot be a torus knot. For any closed 3-manifold $M$, we show that there does not admit any structurally stable non-singular flow $f^t$ whose non-wandering set $NW(f^t)$ consists of a 2-dimensional expanding attractor and a repelling periodic trajectory so that the repelling periodic trajectory is a trivial knot (i.e., it bounds a disk in $M$).

Theorems & Definitions (11)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Lemma 1
• Proposition 1.1
• Proposition 1.2
• Conjecture 1
• Conjecture 2
• Conjecture 3