Diffeomorphisms with Intermingled Attracting Basins

We construct a diffeomorphism $F$ of a manifold with boundary into itself with the following property. The attractor of $F$ has two components, and the attracting basins of these components are dense in the phase space and have positive measure. We prove that the class of examples constructed in the paper has codimension infinity in the space of all diffeomorphisms of the same manifold.