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

2025/10/03 by Z. Lai, В. С. Медведев, Lai, Zhentao +5
Mathematics · Physics and Astronomy · #37D15 #57R50 #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems

paper · pdf · doi:10.48550/arxiv.2510.02704

openalex publication_date 2025/10/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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=\mathbbS³, 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).

Citations

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

Related

• On 2-dimensional expanding attractors of A-flows
• Essential dynamics in chaotic attractors
• Knotted toroidal sets, attractors and incompressible surfaces
• Graph representations of surface flows
• On a structure of non-wandering set of an Ω-stable 3-diffeomorphism possessing a hyperbolic attractor