Simultaneous continuation of infinitely many sinks at homoclinic bifurcations.

We prove that the C 3 diffeomorphisms on surfaces, exhibiting infinitely many sinks near the generic unfolding of a quadratic homoclinic tangency of a dissipative saddle, can be perturbed along an infinite dimensional manifold of C 3 diffeomorphisms such that infinitely many sinks persist simultaneously. On the other hand, if they are perturbed along one-parameter families that unfold generically the quadratic tangencies, then at most a finite number of those sinks have continuation.