Geodesic
Omega-Limit Sets of Generic Points of Partially Hyperbolic Diffeomorphisms
Funkcionalʹnyj analiz i ego priloženiâ, Tome 50 (2016) no. 1, pp. 59-66

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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.
Keywords: attractors, partial hyperbolicity. 
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     author = {S. S. Minkov and A. V. Okunev},
     title = {Omega-Limit {Sets} of {Generic} {Points} of {Partially} {Hyperbolic} {Diffeomorphisms}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
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S. S. Minkov; A. V. Okunev. Omega-Limit Sets of Generic Points of Partially Hyperbolic Diffeomorphisms. Funkcionalʹnyj analiz i ego priloženiâ, Tome 50 (2016) no. 1, pp. 59-66. http://geodesic.mathdoc.fr/item/FAA_2016_50_1_a4/