Functional Analysis and Its Applications

, Volume 46, Issue 4, pp 239–248 | Cite as

Multidimensional bony attractors

  • Yu. S. Ilyashenko


In this paper we study attractors of skew products, for which the following dichotomy is ascertained. These attractors either are not asymptotically stable or possess the following two surprising properties. The intersection of the attractor with some invariant submanifold does not coincide with the attractor of the restriction of the skew product to this submanifold but contains this restriction as a proper subset. Moreover, this intersection is thick on the submanifold, that is, both the intersection and its complement have positive relative measure. Such an intersection is called a bone, and the attractor itself is said to be bony. These attractors are studied in the space of skew products. They have the important property that, on some open subset of the space of skew products, the set of maps with such attractors is, in a certain sense, prevalent, i.e., “big.” It seems plausible that attractors with such properties also form a prevalent subset in an open subset of the space of diffeomorphisms.

Key words

attractor skew product invariant set 


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Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  1. 1.Cornell UniversityIthacaUSA
  2. 2.Independent Moscow University Steklov Mathematical Institute National Research University “Higher School of Economics”Moscow State UniversityMoscowRussia

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