Generation of homoclinic tangencies by $C^1$-perturbations

Given a $C^1$-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 $C^1$-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 $C^1$-perturbation generates a homoclinic tangency related to $P$.