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Funktsional. Anal. i Prilozhen., 2012, Volume 46, Issue 4, Pages 1–13 (Mi faa3088)  

This article is cited in 2 scientific papers (total in 2 papers)

Multidimensional Bony Attractors

Yu. S. Ilyashenkoabcd

a Cornell University
b M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
c Independent University of Moscow
d Steklov Mathematical Institute of the Russian Academy of Sciences

Abstract: 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.

Keywords: attractor, skew product, invariant set

Funding Agency Grant Number
National Science Foundation 0700973
Russian Foundation for Basic Research 10-01-00739_a
10-01-93115_НЦНИЛ_а


DOI: https://doi.org/10.4213/faa3088

Full text: PDF file (186 kB)
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English version:
Functional Analysis and Its Applications, 2012, 46:4, 239–248

Bibliographic databases:

UDC: 517.938
Received: 06.09.2011

Citation: Yu. S. Ilyashenko, “Multidimensional Bony Attractors”, Funktsional. Anal. i Prilozhen., 46:4 (2012), 1–13; Funct. Anal. Appl., 46:4 (2012), 239–248

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. L. J. Diaz, K. Gelfert, “Porcupine-like horseshoes: topological and ergodic aspects”, Progress and challenges in dynamical systems, Springer Proc. Math. Stat., 54, ed. S. Ibanez, J. DelRio, A. Pumarino, J. Rodriguez, Springer, Heidelberg, 2013, 199–219  crossref  mathscinet  zmath  isi  scopus
    2. Yu. Ilyashenko, I. Shilin, “Attractors and skew products”, Modern theory of dynamical systems: a tribute to Dmitry Victorovich Anosov, Contemp. Math., 692, ed. A. Katok, Y. Pesin, F. Hertz, Amer. Math. Soc., Providence, RI, 2017, 155–175  crossref  mathscinet  zmath  isi  scopus
  • Функциональный анализ и его приложения Functional Analysis and Its Applications
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