Simultaneous Continuation of Infinitely Many Sinks Near a Quadratic   Homoclinic Tangency
  We prove that the $C^3$ diffeomorphisms on surfaces, exhibiting infinitely many sinksnear 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. 
Eleonora Catsigeras, Marcelo Cerminara, Heber Enrich