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Uspekhi Mat. Nauk, 2020, Volume 75, Issue 2(452), Pages 3–60 (Mi umn9922)  

This article is cited in 1 scientific paper (total in 1 paper)

Solenoidal attractors of diffeomorphisms of annular sets

S. D. Glyzina, A. Yu. Kolesova, N. Kh. Rozovb

a Demidov Yaroslavl State University
b Lomonosov Moscow State University

Abstract: An arbitrary diffeomorphism $\Pi$ of an annular set of the form $K=B\times \mathbb{T}$ is considered, where $B$ is a ball in a Banach space and $\mathbb{T}$ is a (finite- or infinite-dimensional) torus. A system of effective sufficient conditions is proposed which ensure that $P$ has a global attractor $A=\bigcap_{n\geqslant 0}\Pi^n(K)$ that can be represented as a generalized solenoid, that is, the inverse limit $\mathbb{T}\xleftarrow{G}\mathbb{T}\xleftarrow{G}\cdots\xleftarrow{G}\mathbb{T}\xleftarrow{G}\cdots$, where $G$ is an expanding linear endomorphism of the torus $\mathbb{T}$. Furthermore, the restriction $\Pi|_{A}$ is topologically conjugate to a shift map of the solenoid.
Bibliography: 25 titles.

Keywords: annular set, diffeomorphism, attractor, generalized solenoid, shift map, hyperbolicity.

Funding Agency Grant Number
Russian Foundation for Basic Research 18-29-10055
This research was carried out with the support of the Russian Foundation for Basic Research (grant no. 18-29-10055).


DOI: https://doi.org/10.4213/rm9922

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English version:
Russian Mathematical Surveys, 2020, 75:2, 197–252

Bibliographic databases:

UDC: 517.926
MSC: 37D20
Received: 29.10.2019

Citation: S. D. Glyzin, A. Yu. Kolesov, N. Kh. Rozov, “Solenoidal attractors of diffeomorphisms of annular sets”, Uspekhi Mat. Nauk, 75:2(452) (2020), 3–60; Russian Math. Surveys, 75:2 (2020), 197–252

Citation in format AMSBIB
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\by S.~D.~Glyzin, A.~Yu.~Kolesov, N.~Kh.~Rozov
\paper Solenoidal attractors of diffeomorphisms of annular sets
\jour Uspekhi Mat. Nauk
\yr 2020
\vol 75
\issue 2(452)
\pages 3--60
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\crossref{https://doi.org/10.4213/rm9922}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=4081968}
\elib{https://elibrary.ru/item.asp?id=45311397}
\transl
\jour Russian Math. Surveys
\yr 2020
\vol 75
\issue 2
\pages 197--252
\crossref{https://doi.org/10.1070/RM9922}
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\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85091440940}


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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. S. D. Glyzin, A. Yu. Kolesov, N. Kh. Rozov, “On the Existence and Stability of an Infinite-Dimensional Invariant Torus”, Math. Notes, 109:4 (2021), 534–550  mathnet  crossref  crossref  isi
  • Успехи математических наук Russian Mathematical Surveys
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