A homoclinic tangency of a diffeomorphism $f$ of a smooth manifold $M$ of dimension at least $2$ is an orbit of non-transverse intersection between the stable and unstable manifolds of a saddle periodic point of $f$. Let $\dim M \ge 3$, and let $\{f_t\}$ be any finite-parameter $C^r$ family, $2\le r<\infty$, which has a homoclinic tangency to a weakly dissipative saddle for some parameter value $t$. We show that $\{f_t\}$ lies in the $C^r$-closure of an open set \mathbf{U} in the space of $C^r$ families such that generic families in \mathbf{U} display infinitely many sinks for an open set of parameter values. This is joint work with Dmitry Turaev.
