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

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 consisting of the casing M(Λ a ) and the corresponding fibre link L(Λ a ) is an invariant of conjugacy of the restriction of the flow f t on the basin of the attractor Λ a .