C. A. Morales

Homoclinic classes for generic C^1 vector fields

We prove that homoclinic classes for a residual set of C^1 vector fields X on closed n-manifolds are maximal transitive and depend continuously on periodic orbit data. In addition, X does not exhibit cycles formed by homoclinic classes. We also prove that a homoclinic class of X is isolated if and only if it is Omega-isolated, and that it is the intersection of its stable set and its unstable set. All these properties are well known for structurally stable Axiom A vector fields.

The explosion of singular-hyperbolic attractors

A singular-hyperbolic attractor for vector fields is a partially hyperbolic attractor with singularities (that are hyperbolic) and volume expanding central direction. The geometric Lorenz attractor is the most representative example of a singular-hyperbolic attractor. In this paper, we prove that if

Examples of singular-hyperbolic attracting sets

\Lambda

A dichotomy for higher-dimensional flows

is a singular-hyperbolic attractor of a three-dimensional vector field X , then there is a neighborhood U of

Maximal transitive sets with singularities for genericC1 vector fields

\Lambda

A sectional-Anosov connecting lemma

in M such that every attractor in U of a C r vector field C r close to X is singular, i.e. it contains a singularity. With this result we prove the following corollaries. There are neighborhoods U of

Axiom A flows with a transverse torus

\Lambda

Attractors and orbit-flip homoclinic orbits for star flows

(in M ) and

On the set of expansive measures

\mathcal U

Transverse surfaces and attractors for 3-flows

of X (in the space of C r vector fields) such that if n denotes the number of singularities of X in