Generation of homoclinic tangencies by C¹-perturbations

Given a C¹-diffeomorphism f of a compact manifold, we show that if the stable/unstable dominated splitting along a saddle is weak enough, then there is a small C¹-perturbation that preserves the orbit of the saddle and that generates a homoclinic tangency related to it. Moreover, we show that the perturbation can be performed preserving a homoclinic relation to another saddle. We derive some consequences on homoclinic classes. In particular, if the homoclinic class of a saddle P has no dominated splitting of same index as P, then a C¹-perturbation generates a homoclinic tangency related to P.