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Uspekhi Mat. Nauk, 2013, Volume 68, Issue 1(409), Pages 129–188 (Mi umn9489)  

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

Morse–Smale cascades on 3-manifolds

V. Z. Grines, O. V. Pochinka

Nizhnii Novgorod State University

Abstract: This is a survey of recent (from 2000) results obtained by the authors in collaboration with Russian and foreign colleagues. The major theme of our investigations involves Morse–Smale cascades on orientable 3-manifolds and includes a complete topological classification of them, a determination of the interconnection between their dynamics and the topology of the ambient manifold, a criterion for embeddability in a topological flow, and necessary and sufficient conditions for such cascades to have an energy function.
Bibliography: 76 titles.

Keywords: Morse–Smale diffeomorphism, topological classification, embedding in a flow, energy function.

Funding Agency Grant Number
Russian Foundation for Basic Research 12-01-00672
11-01-12056-офи-м
Ministry of Education and Science of the Russian Federation 11.G34.31.0039
1.1907.2011


DOI: https://doi.org/10.4213/rm9489

Full text: PDF file (1511 kB)
References: PDF file   HTML file

English version:
Russian Mathematical Surveys, 2013, 68:1, 117–173

Bibliographic databases:

UDC: 517.938
MSC: Primary 37D15; Secondary 37C05, 37C15, 37E30, 37C29, 37B25, 57M30
Received: 09.06.2012

Citation: V. Z. Grines, O. V. Pochinka, “Morse–Smale cascades on 3-manifolds”, Uspekhi Mat. Nauk, 68:1(409) (2013), 129–188; Russian Math. Surveys, 68:1 (2013), 117–173

Citation in format AMSBIB
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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. V. Z. Grines, S. H. Kapkaeva, O. V. Pochinka, “A three-colour graph as a complete topological invariant for gradient-like diffeomorphisms of surfaces”, Sb. Math., 205:10 (2014), 1387–1412  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    2. V. Z. Grines, E. A. Gurevich, O. V. Pochinka, “Topological classification of Morse–Smale diffeomorphisms without heteroclinic intersections”, Journal of Mathematical Sciences, 208:1 (2015), 81–90  crossref  zmath  scopus
    3. Yu B., “Behavior 0 Nonsingular Morse Smale Flows on S$^3$”, Discrete and Continuous Dynamical Systems, 36:1 (2016), 509–540  crossref  mathscinet  zmath  isi  scopus
    4. Vyacheslav Z. Grines, Dmitry S. Malyshev, Olga V. Pochinka, Svetlana Kh. Zinina, “Efficient Algorithms for the Recognition of Topologically Conjugate Gradient-like Diffeomorhisms”, Regul. Chaotic Dyn., 21:2 (2016), 189–203  mathnet  crossref  mathscinet
    5. T. M. Mitryakova, O. V. Pochinka, “Necessary and sufficient conditions for the topological conjugacy of 3-diffeomorphisms with heteroclinic tangencies”, Trans. Moscow Math. Soc., 77 (2016), 69–86  mathnet  crossref  elib
    6. V. Z. Grines, T. V. Medvedev, O. V. Pochinka, “Introduction”, Dynamical systems on 2-and 3-manifolds, Developments in Mathematics, 46, Springer, Cham, 2016, 17–26  mathscinet  isi
    7. V. Z. Grines, T. V. Medvedev, O. V. Pochinka, “General properties of the Morse-Smale diffeomorphisms”, Dynamical systems on 2- and 3-manifolds, Developments in Mathematics, 46, Springer, Cham, 2016, 27–55  crossref  mathscinet  isi  scopus
    8. V. Z. Grines, T. V. Medvedev, O. V. Pochinka, “The classification of the gradient-like diffeomorphisms on 3-manifolds”, Dynamical systems on 2- and 3-manifolds, Developments in Mathematics, 46, Springer, Cham, 2016, 109–118  crossref  mathscinet  isi  scopus
    9. 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
    10. V. Z. Grines, E. Ya. Gurevich, V. S. Medvedev, O. V. Pochinka, “An Analog of Smale's Theorem for Homeomorphisms with Regular Dynamics”, Math. Notes, 102:4 (2017), 569–574  mathnet  crossref  crossref  mathscinet  isi  elib
    11. 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
    12. V. Z. Grines, O. V. Pochinka, “Topological classification of global magnetic fields in the solar corona”, Dynam. Syst., 33:3 (2018), 536–546  crossref  mathscinet  zmath  isi  scopus
    13. V. Z. Grines, E. Ya. Gurevich, O. V. Pochinka, “A Combinatorial Invariant of Morse–Smale Diffeomorphisms without Heteroclinic Intersections on the Sphere $S^n$, $n\ge 4$”, Math. Notes, 105:1 (2019), 132–136  mathnet  crossref  crossref  isi  elib
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