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Tr. Mat. Inst. Steklova, 2010, Volume 271, Pages 111–133 (Mi tm3236)  

This article is cited in 19 scientific papers (total in 19 papers)

Global attractor and repeller of Morse–Smale diffeomorphisms

V. Z. Grinesa, E. V. Zhuzhomab, V. S. Medvedevc, O. V. Pochinkaa

a Lobachevsky State University of Nizhni Novgorod, Nizhni Novgorod, Russia
b Nizhni Novgorod State Pedagogical University, Nizhni Novgorod, Russia
c Research Institute for Applied Mathematics and Cybernetics, Lobachevsky State University of Nizhni Novgorod, Nizhni Novgorod, Russia

Abstract: Let $f$ be an orientation-preserving Morse–Smale diffeomorphism of an $n$-dimensional ($n\ge3$) closed orientable manifold $M^n$. We show the possibility of representing the dynamics of $f$ in a “source–sink” form. The roles of the “source” and “sink” are played by invariant closed sets one of which, $A_f$, is an attractor, and the other, $R_f$, is a repeller. Such a representation reveals new topological invariants that describe the embedding (possibly, wild) of stable and unstable manifolds of saddle periodic points in the ambient manifold. These invariants have allowed us to obtain a classification of substantial classes of Morse–Smale diffeomorphisms on 3-manifolds. In this paper, for any $n\ge3$, we describe the topological structure of the sets $A_f$ and $R_f$ and of the space of orbits that belong to the set $M^n\setminus(A_f\cup R_f)$.

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English version:
Proceedings of the Steklov Institute of Mathematics, 2010, 271, 103–124

Bibliographic databases:

UDC: 517.938
Received in January 2010

Citation: V. Z. Grines, E. V. Zhuzhoma, V. S. Medvedev, O. V. Pochinka, “Global attractor and repeller of Morse–Smale diffeomorphisms”, Differential equations and topology. II, Collected papers. In commemoration of the centenary of the birth of Academician Lev Semenovich Pontryagin, Tr. Mat. Inst. Steklova, 271, MAIK Nauka/Interperiodica, Moscow, 2010, 111–133; Proc. Steklov Inst. Math., 271 (2010), 103–124

Citation in format AMSBIB
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\paper Global attractor and repeller of Morse--Smale diffeomorphisms
\inbook Differential equations and topology.~II
\bookinfo Collected papers. In commemoration of the centenary of the birth of Academician Lev Semenovich Pontryagin
\serial Tr. Mat. Inst. Steklova
\yr 2010
\vol 271
\pages 111--133
\publ MAIK Nauka/Interperiodica
\publaddr Moscow
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. O. V. Pochinka, “Neobkhodimye i dostatochnye usloviya topologicheskoi sopryazhennosti kaskadov Morsa–Smeila na 3-mnogoobraziyakh”, Nelineinaya dinam., 7:2 (2011), 227–238  mathnet  elib
    2. E. V. Zhuzhoma, V. S. Medvedev, “Morse–Smale Diffeomorphisms with Three Fixed Points”, Math. Notes, 92:4 (2012), 497–512  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    3. Medvedev V.S., Zhuzhoma E.V., “Locally Flat and Wildly Embedded Separatrices in Simplest Morse-Smale Systems”, J. Dyn. Control Syst., 18:3 (2012), 433–448  crossref  mathscinet  zmath  isi  elib  scopus
    4. Medvedev V.S., Zhuzhoma E.V., “Morse-Smale Systems with Few Non-Wandering Points”, Topology Appl., 160:3 (2013), 498–507  crossref  mathscinet  zmath  isi  elib  scopus
    5. V. Z. Grines, O. V. Pochinka, “On the Simple Isotopy Class of a Source–Sink Diffeomorphism on the $3$-Sphere”, Math. Notes, 94:6 (2013), 862–875  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    6. V. Z. Grines, E. Ya. Gurevich, E. V. Zhuzhoma, S. Kh. Zinina, “Geteroklinicheskie krivye diffeomorfizmov Morsa–Smeila i separatory v magnitnom pole plazmy”, Nelineinaya dinam., 10:4 (2014), 427–438  mathnet
    7. Grines V. Pochinka O. Zhuzhoma E., “on Families of Diffeomorphisms With Bifurcations of Attractive and Repelling Sets”, Int. J. Bifurcation Chaos, 24:8 (2014), 1440015  crossref  mathscinet  zmath  isi  elib  scopus
    8. Grines V., Medvedev T., Pochinka O., Zhuzhoma E., “on Heteroclinic Separators of Magnetic Fields in Electrically Conducting Fluids”, Physica D, 294 (2015), 1–5  crossref  mathscinet  zmath  isi  scopus
    9. V. Z. Grines, E. V. Zhuzhoma, O. V. Pochinka, “Sistemy Morsa–Smeila i topologicheskaya struktura nesuschikh mnogoobrazii”, Trudy Krymskoi osennei matematicheskoi shkoly-simpoziuma, SMFN, 61, RUDN, M., 2016, 5–40  mathnet
    10. V. Z. Grines, E. Ya. Gurevich, O. V. Pochinka, “On embedding Morse–Smale diffeomorphisms on the sphere in topological flows”, Russian Math. Surveys, 71:6 (2016), 1146–1148  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    11. Grines V.Z. Medvedev T.V. Pochinka O.V., “General Properties of the Morse-Smale Diffeomorphisms”: Grines, VZ Medvedev, TV Pochinka, OV, Dynamical Systems on 2- and 3-Manifolds, Developments in Mathematics, 46, Springer International Publishing Ag, 2016, 27–55  crossref  mathscinet  isi  scopus
    12. Grines V.Z. Medvedev T.V. Pochinka O.V., “The Classification of the Gradient-Like Diffeomorphisms on 3-Manifolds”: Grines, VZ Medvedev, TV Pochinka, OV, Dynamical Systems on 2- and 3-Manifolds, Developments in Mathematics, 46, Springer International Publishing Ag, 2016, 109–118  crossref  mathscinet  isi  scopus
    13. V. Z. Grines, E. V. Zhuzhoma, V. S. Medvedev, “On the structure of the ambient manifold for Morse–Smale systems without heteroclinic intersections”, Proc. Steklov Inst. Math., 297 (2017), 179–187  mathnet  crossref  crossref  mathscinet  isi  elib
    14. Ch. Bonatti, V. Z. Grines, O. V. Pochinka, “Realization of Morse–Smale diffeomorphisms on $3$-manifolds”, Proc. Steklov Inst. Math., 297 (2017), 35–49  mathnet  crossref  crossref  mathscinet  isi  elib
    15. 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
    16. E. V. Zhuzhoma, V. S. Medvedev, N. V. Isaenkova, “O topologicheskoi strukture magnitnogo polya oblastei fotosfery”, Nelineinaya dinam., 13:3 (2017), 399–412  mathnet  crossref  elib
    17. V. Z. Grines, E. V. Zhuzhoma, O. V. Pochinka, “Dinamicheskie sistemy i topologiya magnitnykh polei v provodyaschei srede”, Differentsialnye i funktsionalno-differentsialnye uravneniya, SMFN, 63, no. 3, Rossiiskii universitet druzhby narodov, M., 2017, 455–474  mathnet  crossref
    18. E. V. Zhuzhoma, V. S. Medvedev, “Conjugacy of Morse–Smale Diffeomorphisms with Three Nonwandering Points”, Math. Notes, 104:5 (2018), 753–757  mathnet  crossref  crossref  isi  elib
    19. V. Z. Grines, E. Ya. Gurevich, E. V. Zhuzhoma, O. V. Pochinka, “Classification of Morse–Smale systems and topological structure of the underlying manifolds”, Russian Math. Surveys, 74:1 (2019), 37–110  mathnet  crossref  crossref  adsnasa  isi  elib
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