Omega-Limit Sets of Generic Points of Partially Hyperbolic Diffeomorphisms

We prove that, for any $E^u \oplus E^{cs}$ partially hyperbolic $C^2$ diffeomorphism, the $\omega$-limit set of a generic (with respect to the Lebesgue measure) point is a union of unstable leaves. As a corollary, we prove a conjecture made by Ilyashenko in his 2011 paper that the Milnor attractor is a union of unstable leaves. In the paper mentioned above, Ilyashenko reduced the local generecity of the existence of a “thick” Milnor attractor in the class of boundary-preserving diffeomorphisms of the product of the interval and the 2-torus to this conjecture.