Díaz–Gorodetski homoclinic-class non-hyperbolic measure conjecture

Let $M$ be a compact smooth manifold without boundary, and let $\operatorname{Diff}^1(M)$ be the space of $C^1$ diffeomorphisms of $M$. For a periodic point $p$, its homoclinic class $H(p)$ is the closure of the transverse intersection points between the stable and unstable manifolds of the orbit of $p$. An invariant measure is non-hyperbolic if it has a vanishing Lyapunov exponent. Díaz–Gorodetski's conjecture. For generic $f\in\operatorname{Diff}^1(M)$, every homoclinic class either is uniformly hyperbolic or supports an ergodic non-hyperbolic invariant measure. The paper proves this local conjecture for non-hyperbolic homoclinic classes of generic diffeomorphisms, while noting that the genericity assumption cannot simply be removed.