Two-dimensional attractors of A-flows and fibred links on three-manifolds. (5th May 2022)

• Record Type: Journal Article
• Authors: Medvedev, V; Zhuzhoma, E
• Is Part Of: Nonlinearity. Volume 35:Number 5(2022)
• Page Start: 2192
• Page End: 2205
• Publication Date: 2022-05-05
• Subjects: attractor -- A-flow -- fibred link; 37D20 -- 37C70 -- 37C15 -- 58C30; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/1361-6544/ac5a64
• Languages: English
• ISSNs: 0951-7715

Abstract

Abstract: Let f t be a flow satisfying Smale's Axiom A (in short, A-flow) on a closed orientable three-manifold M 3, and Ω a two-dimensional basic set of f t . First, we prove that Ω is either an expanding attractor or contracting repeller. Next, one considers an A-flow f t with a two-dimensional non-mixing attractor Λ a . We construct a casing M (Λ a ) of Λ a that is a special compactification of the basin of Λ a by a collection of circles L (Λ a ) = { l 1, …, l k } such that M (Λ a ) is a closed three-manifold and L (Λ a ) is a fibre link in M (Λ a ). In addition, f t is extended on M (Λ a ) to a nonsingular structurally stable flow with the non-wandering set consisting of the attractor Λ a and the repelling periodic trajectories l 1, …, l k . We show that if a closed orientable three-manifold M 3 has a fibred link L = { l 1, …, 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, …, l k . This allows us to prove that any closed orientable n -manifold, n ⩾ 3, admits an A-flow with a two-dimensional attractor. We prove that the pair M ( Λ a ) ; L ( Λ a ) consisting of the casing M (Λ a ) and the corresponding fibre link L (Λ a ) is an invariant of conjugacy of the restriction f t | W s ( Λ a ) of the flow f t on the basin of the attractor Λ a .