In this talk we shall study continuations of attractors of embeddings in manifolds. We shall introduce the abstract basin of attraction of such attractors and we shall see that if two attractors are related by continuation, their abstract basins are homeomorphic. Moreover, if this continuation is achieved by a small perturbation, then, the homeomorphism can be chosen to be the identity close to the original attractor. We shall make use of this rigidity property in order to characterize the Cech cohomology of the attractors expelled after a generalized Hopf bifurcation of an attractor.

These results have been obtained in collaboration with J.J. Sánchez-Gabites