Abstract
We prove that, for any E u ⊕ E cs partially hyperbolic C 2 diffeomorphism, the ω-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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
V. I. Arnold, V. S. Afraimovich, Y. S. Ilyashenko, and L. P. Shilnikov, “Bifurcation theory”, in: Itogi Nauki i Tekhniki. Sovremennye Problemy Matematiki. Fundamental’nye Napravleniya, VINITI, Moscow, 1986.
L. Barreira and Ya. B. Pesin, Nonuniform Hyperbolicity. Dynamics of Systems with Nonzero Lyapunov Exponents, Cambridge University Press, Cambridge, 2007.
C. Bonatti, L. J. Diaz, and M. Viana, Dynamics Beyond Uniform Hyperbolicity. A global geometric and probabilistic perspective, Springer-Verlag, Berlin, 2005.
Yu. S. Ilyashenko, “Thick attractors of boundary preserving diffeomorphisms,” Indag. Math., 22:3–4 (2011), 257–314.
Yu. G. Kudryashov, Bony attractors in higher dimensions, http://arxiv.org/abs/1305.3889v1.
J. Milnor, “Fubini foiled: Katok’s paradoxical example in measure theory,” Math. Intelligencer, 19:2 (1997), 30–32.
C. A. Morales and M. J. Pacifico, “Lyapunov stability of ?-limit sets,” Discrete Contin. Dyn. Syst., Ser. A, 8:3 (2002), 671–674.
Ya. Pesin, Lectures on Partial Hyperbolicity and Stable Ergodicity, European Mathematical Society, Zürich, 2004.
C. Pugh, M. Viana, and A. Wilkinson, Absolute continuity of foliations, http://w3.impa.br/~viana/out/pvw.pdf.
M. Shub and A. Wilkinson, “Pathological foliations and removable zero exponents,” Invent. Math., 139:3 (2000), 495–508.
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported in part by the Simons Foundation and by RFBR grant 16-01-00748-a.
Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 50, No. 1, pp. 59–66, 2016
Rights and permissions
About this article
Cite this article
Minkov, S.S., Okunev, A.V. Omega-Limit Sets of Generic Points of Partially Hyperbolic Diffeomorphisms. Funct Anal Its Appl 50, 48–53 (2016). https://doi.org/10.1007/s10688-016-0127-2
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10688-016-0127-2