Homoclinic Bifurcations and Persistence of Nonuniformly Hyperbolic Attractors

Let ϕ: M → M be a general smooth transformation on a riemannian manifold. A main object of study in dynamics is the asymptotic behavior of the orbits ϕ n (z) = ϕ○…○ϕ(z ∈ M, as time n goes to infinity. Typical forms of behavior — occurring for “many” z ∈ M — are, of course, of particular relevance and this leads us to the notion of attractor. By an attractor we mean a (compact) ϕ-invariant set Λ ⊂ M that is dynamically invisible and whose basin — the set of points z ∈ M for which ϕ n (z) → Λ as n → + ∞ — is a large set. Dynamical invisibility can be expressed by the existence os a dense orbit in Λ (if Λ supports a “natural” ϕ-invariant measure, one may also require that ϕ be ergodic with respect to such a measure). As for the basin, it must have positive Lebesgue volume or, even, nonempty interior; in all the cases we will consider here the basin actually contains a full neighborhood of the attractor.