Turaev's Newhouse-region conjecture for homoclinic tangencies

Let $r\geq 2$ and let a homoclinic tangency be given. A Newhouse region is an open set of diffeomorphisms in which a residual subset exhibits infinitely many sinks or sources. A tangency is volume-hyperbolic when its invariant splitting has uniformly contracted volume on the stable extremal bundle and uniformly expanded volume on the unstable extremal bundle.

Turaev's conjecture. There is a Newhouse region $C^r$-close to a homoclinic tangency if and only if that tangency is not volume-hyperbolic.

The conjecture asserts that volume-hyperbolicity is the only obstruction to Newhouse phenomena near a tangency. The paper states that its results prove this dichotomy, so the conjecture is solved in the setting considered there.