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Regular and Chaotic Dynamics, 2006, Volume 11, Issue 2, Pages 225–246
DOI: https://doi.org/10.1070/RD2006v011n02ABEH000347
(Mi rcd670)
 

This article is cited in 7 scientific papers (total in 7 papers)

On the 70th birthday of L.P. Shilnikov

Expanding attractors

V. Z. Grinesa, E. V. Zhuzhomab

a Department of Mathematics, Agriculture Academy of Nizhny Novgorod, 97 Gagarin str., 603107 Nizhny Novgorod, Russia
b Department of Applied Mathematics, Nizhny Novgorod State Technical University, 24 Minin str., 603600 Nizhny Novgorod, Russia
Citations (7)
Abstract: The article is a survey on local and global structures (including classification results) of expanding attractors of diffeomorphisms f:MM of a closed smooth manifold M. Beginning with the most familiar expanding attractors (Smale solenoid; DA-attractor; Plykin attractor; Robinson–Williams attractors), one reviews the Williams theory, Bothe's classification of one-dimensional solenoids in 3-manifolds, Grines–Plykin–Zhirov's classification of one-dimensional expanding attractors on surfaces, and Grines–Zhuzhoma's classification of codimension one expanding attractors of structurally stable diffeomorphisms. The main theorems are endowed with ideas of proof.
Keywords: Axiom A diffeomorphisms, (codimension one) expanding attractors, structurally stable diffeomorphisms, hyperbolic automorphisms.
Received: 28.05.2005
Accepted: 15.11.2005
Bibliographic databases:
Document Type: Article
MSC: 37D20, 37C70, 37C15
Language: English
Citation: V. Z. Grines, E. V. Zhuzhoma, “Expanding attractors”, Regul. Chaotic Dyn., 11:2 (2006), 225–246
Citation in format AMSBIB
\Bibitem{GriZhu06}
\by V. Z. Grines, E. V. Zhuzhoma
\paper Expanding attractors
\jour Regul. Chaotic Dyn.
\yr 2006
\vol 11
\issue 2
\pages 225--246
\mathnet{http://mi.mathnet.ru/rcd670}
\crossref{https://doi.org/10.1070/RD2006v011n02ABEH000347}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=2245079}
\zmath{https://zbmath.org/?q=an:1164.37317}
Linking options:
  • https://www.mathnet.ru/eng/rcd670
  • https://www.mathnet.ru/eng/rcd/v11/i2/p225
  • This publication is cited in the following 7 articles:
    1. Vladislav S. Medvedev, Evgeny V. Zhuzhoma, “On the Existence of Expanding Attractors with Different Dimensions”, Regul. Chaotic Dyn., 30:1 (2025), 93–102  mathnet  crossref
    2. E. V. Zhuzhoma, V. S. Medvedev, “On Expanding Attractors of Arbitrary Codimension”, J Math Sci, 2025  crossref
    3. E. V. Zhuzhoma, V. S. Medvedev, “O rastyagivayuschikhsya attraktorakh proizvolnoi korazmernosti”, SMFN, 70, no. 3, Rossiiskii universitet druzhby narodov, M., 2024, 389–402  mathnet  crossref
    4. V. Z. Grines, O. V. Pochinka, S. van Strien, “On 2-diffeomorphisms with one-dimensional basic sets and a finite number of moduli”, Mosc. Math. J., 16:4 (2016), 727–749  mathnet  crossref  mathscinet
    5. Sergey P. Kuznetsov, Hyperbolic Chaos, 2012, 3  crossref
    6. S. P. Kuznetsov, “Dynamical chaos and uniformly hyperbolic attractors: from mathematics to physics”, Phys. Usp., 54:2 (2011), 119–144  mathnet  crossref  crossref  adsnasa  isi  elib
    7. E. V. Zhuzhoma, V. S. Medvedev, “Surface Basic Sets with Wildly Embedded Supporting Surfaces”, Math. Notes, 85:3 (2009), 353–365  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
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