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arXiv · 2510.02704

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

Zhentao Lai·V. Medvedev·Bin Yu·E. Zhuzhoma
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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 $M$).

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Zhentao Lai, V. Medvedev, Bin Yu, E. Zhuzhoma. 2025-10-03. Nonsingular structural stable chaotic 3-flows of attractor-repeller type. https://arxiv.org/abs/2510.02704

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