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Mat. Zametki, 2005, Volume 78, Issue 6, Pages 813–826 (Mi mz2655)  

This article is cited in 13 scientific papers (total in 13 papers)

On Surface Attractors and Repellers in 3-Manifolds

V. Z. Grinesa, V. S. Medvedevb, E. V. Zhuzhomac

a Nizhnii Novgorod State Agricultural Academy
b Research Institute for Applied Mathematics and Cybernetics, N. I. Lobachevski State University of Nizhnii Novgorod
c Nizhny Novgorod State Technical University

Abstract: We show that if $f\colon M^3\to M^3$ is an $A$ diffeomorphism with a surface two-dimensional attractor or repeller $\mathscr B$ with support $M^2_{\mathscr B}$, then $\mathscr B=M^2_{\mathscr B}$ and there exists a $k\ge1$ such that
  • 1) $M^2_{\mathscr B}$ is the disjoint union $M^2_1\cup…\cup M^2_k$ of tame surfaces such that each surface $M^2_i$ is homeomorphic to the 2-torus $T^2$;
  • 2) the restriction of $f^k$ to $M^2_i$, $i\in\{1,…,k\}$, is conjugate to an Anosov diffeomorphism of the torus $T^2$.


DOI: https://doi.org/10.4213/mzm2655

Full text: PDF file (293 kB)
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English version:
Mathematical Notes, 2005, 78:6, 757–767

Bibliographic databases:

UDC: 513.83+517.9
Received: 07.02.2005

Citation: V. Z. Grines, V. S. Medvedev, E. V. Zhuzhoma, “On Surface Attractors and Repellers in 3-Manifolds”, Mat. Zametki, 78:6 (2005), 813–826; Math. Notes, 78:6 (2005), 757–767

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. Grines V. Z., Zhuzhoma E. V., “Expanding attractors”, Regul. Chaotic Dyn., 11:2 (2006), 225–246  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    2. 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
    3. Brown A.W., “Nonexpanding attractors: conjugacy to algebraic models and classification in 3-manifolds”, J. Mod. Dyn., 4:3 (2010), 517–548  crossref  mathscinet  zmath  isi  scopus  scopus
    4. Grines V.Z., Levchenko Yu.A., “On a Topological Classification of Diffeomorphisms on 3-Manifolds with Two-Dimensional Surface Attractors and Repellers”, Dokl. Math., 86:3 (2012), 747–749  crossref  mathscinet  zmath  zmath  isi  elib  scopus
    5. V. Z. Grines, Yu. A. Levchenko, O. V. Pochinka, “O topologicheskoi klassifikatsii diffeomorfizmov na 3-mnogoobraziyakh s poverkhnostnymi dvumernymi attraktorami i repellerami”, Nelineinaya dinam., 10:1 (2014), 17–33  mathnet
    6. Vyacheslav Z. Grines, Yulia A. Levchenko, Vladislav S. Medvedev, Olga V. Pochinka, “On the Dynamical Coherence of Structurally Stable 3-diffeomorphisms”, Regul. Chaotic Dyn., 19:4 (2014), 506–512  mathnet  crossref  mathscinet  zmath
    7. V. Z. Grines, Yu. A. Levchenko, O. V. Pochinka, “Topological Classification of Structurally Stable 3-Diffeomorphisms with Two-Dimensional Basis Sets”, Math. Notes, 97:2 (2015), 304–306  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    8. V. Z. Grines, Ye. V. Zhuzhoma, O. V. Pochinka, “Rough diffeomorphisms with basic sets of codimension one”, Journal of Mathematical Sciences, 225:2 (2017), 195–219  mathnet  crossref
    9. Grines V., Levchenko Yu., Medvedev V., Pochinka O., “the Topological Classification of Structurally Stable 3-Diffeomorphisms With Two-Dimensional Basic Sets”, Nonlinearity, 28:11 (2015), 4081–4102  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    10. Grines V.Z., Medvedev T.V., Pochinka O.V., “Dynamical Systems on 2-and 3-Manifolds Introduction”: Grines, VZ Medvedev, TV Pochinka, OV, Dynamical Systems on 2- and 3-Manifolds, Developments in Mathematics, 46, Springer International Publishing Ag, 2016, XVII–XXVI  mathscinet  isi
    11. Vyacheslav Z. Grines, Elena Ya. Gurevich, Olga V. Pochinka, “On the Number of Heteroclinic Curves of Diffeomorphisms with Surface Dynamics”, Regul. Chaotic Dyn., 22:2 (2017), 122–135  mathnet  crossref
    12. V. Z. Grines, O. V. Pochinka, “Postroenie energeticheskikh funktsii dlya $\Omega$-ustoichivykh diffeomorfizmov na $2$- i $3$-mnogoobraziyakh”, Trudy Krymskoi osennei matematicheskoi shkoly-simpoziuma, SMFN, 63, no. 2, Rossiiskii universitet druzhby narodov, M., 2017, 191–222  mathnet  crossref
    13. Grines V. Zhuzhoma E., “Around Anosov-Weil Theory”, Modern Theory of Dynamical Systems: a Tribute to Dmitry Victorovich Anosov, Contemporary Mathematics, 692, ed. Katok A. Pesin Y. Hertz F., Amer Mathematical Soc, 2017, 123–154  crossref  mathscinet  zmath  isi  scopus
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