Pugh–Shub accessibility conjecture for partially hyperbolic diffeomorphisms

Let $M$ be a closed Riemannian manifold, and let $f:M\to M$ be a partially hyperbolic diffeomorphism, meaning that $TM$ admits a $Df$-invariant splitting

$$TM=E^s\oplus E^c\oplus E^u,$$

where $E^s$ is uniformly contracting, $E^u$ is uniformly expanding, and $E^c$ is dominated by the other two subbundles. The diffeomorphism $f$ is accessible if any two points of $M$ can be joined by a piecewise smooth path tangent to $E^s$ or $E^u$. Pugh–Shub accessibility conjecture. Accessibility holds for an open and dense subset of partially hyperbolic diffeomorphisms. This is a central conjecture in partially hyperbolic dynamics and was proposed as part of the Stable Ergodicity Conjecture; the supplied text does not state whether it has been resolved.