Two-dimensional attractors of A-flows and fibred links on three-manifolds

Nonlinearity. 2022. Vol. 35. No. 5. P. 2192–2205.

Medvedev V., Zhuzhoma E. V.

Let $f^t$ be a flow satisfying Smale's Axiom A (in short, A-flow) on a closed orientable 3-manifold $M^3$, and $\Omega$ a two-dimensional basic set of $f^t$.
First, we prove that $\Omega$ is either an expanding attractor or contracting repeller. Next, one considers an A-flow $f^t$ with a two-dimensional non-mixing attractor $\Lambda_a$.
We construct a casing $M(\Lambda_a)$ of $\Lambda_a$ that is a special compactification of the basin of $\Lambda_a$ by a collection of circles $L(\Lambda_a)=\{l_1,\ldots,l_k\}$ such that $M(\Lambda_a)$ is a closed 3-manifold and $L(\Lambda_a)$ is a fiber link in $M(\Lambda_a)$. In addition, $f^t$ is extended on $M(\Lambda_a)$ to a nonsingular structurally stable flow with the non-wandering set consisting of the attractor $\Lambda_a$ and the repelling periodic trajectories $l_1$, $\ldots$, $l_k$. We show that if a closed orientable 3-manifold $M^3$ has a fibered link $L=\{l_1,\ldots,l_k\}$ then $M^3$ admits an A-flow $f^t$ with the non-wandering set containing a two-dimensional non-mixing attractor and the repelling isolated periodic trajectories $l_1$, $\ldots$, $l_k$. This allows us to prove that any closed orientable $n$-manifold, $n\geq 3$, admits an A-flow with a two-dimensional attractor. We prove that the pair $\left(M(\Lambda_a);L(\Lambda_a)\right)$ consisting of the casing $M(\Lambda_a)$ and the corresponding fiber link $L(\Lambda_a)$ is an invariant of conjugacy of the restriction $f^t|_{W^s(\Lambda_a)}$ of the flow $f^t$ on the basin of the attractor $\Lambda_a$.

Keywords: attractor, A-flow, fibered link

Publication based on the results of: Теория динамических систем и ее приложения (2023)