Generic bi-Lyapunov stable homoclinic classes

We study, for C1 generic diffeomorphisms, homoclinic classes which are Lyapunov stable both for backward and forward iterations. We prove that they must admit a dominated splitting and show that under some hypothesis they must be the whole manifold. As a consequence of our results we also prove that in dimension 2 the class must be the whole manifold and in dimension 3, these classes must have nonempty interior. Many results on Lyapunov stable homoclinic classes for C1-generic diffeomorphisms are also deduced