Attractors: Persistence, and Density of Their Basins

Abstract

Abstract. An investigation of qualitative features of flows on manifolds, in terms of their attractors and quasi-attractors. A quasi-attractor is any nonempty intersection of attractors. It is shown that quasi-attractors other than attractors occur for a large set of flows. It is also shown that for a generic flow (for each flow in a residual subset of the set of all flows), each attractor "persists" as an attractor of all nearby flows. Similar statements are shown to hold with "quasi-attractor", "chain transitive attractor", and "chain transitive quasi-attractor" in place of "attractor". Finally, the set of flows under which almost all points tend asymptotically to a chain transitive quasi-attractor is characterized in terms of stable sets of invariant sets.0. Introduction. This paper investigates qualitative properties of attractors of flows on compact manifolds. An attractor of a flow, /, is a nonempty, compact, /-invariant set A that has a neighborhood, U, satisfying C\t>Qf(t, U) = A. The basin of attraction of the attractor A is the set of all points in the manifold that approach A asymptotically under the forward flow.The central concern is to determine when there is a collection of attractors of a given flow such that:(a) Each attractor in the collection is "indecomposable" in some sense.(b) Each of these attractors is "persistent" under perturbations of the flow.(c) The union of the basins of these attractors forms a dense subset in the manifold.The origin of the problem is a conjecture by R. Thorn: Conjecture [T, p. 39]. Given r > 1, there is a set of flows, S, residual in the set of all C flows on M, such that / £ S implies that / has a finite number of structurally stable, topologically transitive attractors whose basins are dense in M. Note that the attractors Thorn is talking about satisfy strong forms of the notions of "indecomposability" (topological transitivity) and "persistence" (structural stability). The primary examples of the exact situation Thom describes are the Axiom A and no cycles flows (see [S] or [N] for definitions), except of course that these flows do not in general form a residual subset. In the time since the original conjecture was made, a number of examples have shown that the conjecture needs

Perturbing a product of stable flows

“…There is earlier related work of Hurley [8] and Moreva [11]. They found a generic stability property for attractors: a perturbation of any flow g in a certain residual set has some attractor close to an attractor of g. This is stronger than our results in that no assumption like axiom A is made.…”

Robust transitive singular sets for 3-flows are partially hyperbolic attractors or repellers

“…Since t n → ∞ as n → ∞, we can assume that t n > T γ for every n. The continuity of the splitting E s ⊕ E cu over T ΛX (U ) M with the flow together with (16) give, for n big enough, that…”

Bifurcation and chain recurrence

“…Recall that a subset S of a topological space X is residual if S can be realized as a The relevant facts are that (Diff 1 (M), d\) is a Baire space [2], so that any residual subset is dense; and that the metric topology makes (FM, d H ) a compact metric space [4]. A more detailed description is contained in [3]. The proof of (a) is essentially due to C. Conley.…”