Bonatti–Díaz hyperbolicity conjecture

Let $M$ be a compact manifold and let $\operatorname{Diff}^1(M)$ be the space of $C^1$ diffeomorphisms. A diffeomorphism is hyperbolic if its nonwandering dynamics has a hyperbolic splitting, and a robust heterodimensional cycle is a heterodimensional cycle that persists under small perturbations. Bonatti–Díaz hyperbolicity conjecture. Any diffeomorphism can be approximated in $\operatorname{Diff}^1(M)$ by one which is hyperbolic or exhibits a robust heterodimensional cycle. This is motivated by the observation that known examples of $C^1$-generic nonhyperbolic systems involve heterodimensional cycles; the source gives no resolution.