Singular Continuations of Attractors

We study dynamical and topological properties of the singularities of continuations of attractors of flows on manifolds. Despite the fact that these singularities are not isolated invariant sets, they share many of the properties of attractors; in particular, they have finitely generated Cech homology and cohomology, and they have the Cech homotopy type of attractors. This means that, from a global point of view, the singularities of continuations are topological objects closely related to finite polyhedra. The global structure is preserved even for weaker forms of continuation. An interesting case occurs with the Lorenz system for parameter values close to the situation of preturbulence. A general result, motivated by this particular case, is presented.