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

Let f^t be a flow satisfying Smale's Axiom A (in short, A-flow) on a closed orientable three-manifold M³, 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) = {l₁, ..., 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 l₁, ..., l_k. We show that if a closed orientable three-manifold M³ has a fibred link L = {l₁, ..., l_k} then M³ admits an A-flow f^t with the non-wandering set containing a two-dimensional non-mixing attractor and the repelling isolated periodic trajectories l₁, ..., 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 $\left(M({{\Lambda}}_{a});L({{\Lambda}}_{a})\right)$ consisting of the casing M(Λ_a) and the corresponding fibre link L(Λ_a) is an invariant of conjugacy of the restriction ${f}^{t}{\vert }_{{W}^{s}({{\Lambda}}_{a})}$ of the flow f^t on the basin of the attractor Λ_a.