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Mat. Sb. (N.S.), 1973, Volume 90(132), Number 3, Pages 372–402 (Mi msb3055)  

This article is cited in 35 scientific papers (total in 35 papers)

On some invariants of dynamical systems on two-dimensional manifolds (necessary and sufficient conditions for the topological equivalence of transitive dynamical systems)

S. Kh. Aranson, V. Z. Grines


Abstract: In this paper, topological invariants of dynamical systems given on a two-dimensional manifold $M^2$ of genus $p>1$ are selected which allow one to distinguish topologically inequivalent systems which have nonclosed, Poisson stable trajectories and non-null-homotopic closed trajectories.
A necessary and sufficient condition for the topological equivalence of transitive dynamical systems on $M^2$ is established.
Figures: 6.
Bibliography: 20 titles.

Full text: PDF file (3720 kB)
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English version:
Mathematics of the USSR-Sbornik, 1973, 19:3, 365–393

Bibliographic databases:

UDC: 517.917
MSC: 34C35, 58F10
Received: 22.06.1971

Citation: S. Kh. Aranson, V. Z. Grines, “On some invariants of dynamical systems on two-dimensional manifolds (necessary and sufficient conditions for the topological equivalence of transitive dynamical systems)”, Mat. Sb. (N.S.), 90(132):3 (1973), 372–402; Math. USSR-Sb., 19:3 (1973), 365–393

Citation in format AMSBIB
\Bibitem{AraGri73}
\by S.~Kh.~Aranson, V.~Z.~Grines
\paper On~some invariants of dynamical systems on two-dimensional manifolds (necessary and sufficient conditions for the topological equivalence of transitive dynamical systems)
\jour Mat. Sb. (N.S.)
\yr 1973
\vol 90(132)
\issue 3
\pages 372--402
\mathnet{http://mi.mathnet.ru/msb3055}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=339275}
\zmath{https://zbmath.org/?q=an:0252.54027}
\transl
\jour Math. USSR-Sb.
\yr 1973
\vol 19
\issue 3
\pages 365--393
\crossref{https://doi.org/10.1070/SM1973v019n03ABEH001784}


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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. S. Kh. Aranson, V. Z. Grines, “O topologicheskoi ekvivalentnosti minimalnykh mnozhestv dinamicheskikh sistem na dvumernykh mnogoobraziyakh”, UMN, 28:4(172) (1973), 205–206  mathnet  mathscinet  zmath
    2. V. Z. Grines, “O topologicheskoi ekvivalentnosti odnomernykh bazisnykh mnozhestv diffeomorfizmov na dvumernykh mnogoobraziyakh”, UMN, 29:6(180) (1974), 163–164  mathnet  mathscinet  zmath
    3. S. Kh. Aranson, V. Z. Grines, “On the representation of minimal sets of currents on two-dimensional manifolds by geodesics”, Math. USSR-Izv., 12:1 (1978), 103–124  mathnet  crossref  mathscinet  zmath
    4. Gilbert Levitt, “Foliations and laminations on hyperbolic surfaces”, Topology, 22:2 (1983), 119  crossref
    5. R. V. Plykin, “On the geometry of hyperbolic attractors of smooth cascades”, Russian Math. Surveys, 39:6 (1984), 85–131  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    6. S. Kh. Aranson, “Topological equivalence of fiberings with singularities and homeomorphisms with invariant fiberings on two-dimensional manifolds”, Russian Math. Surveys, 41:3 (1986), 197–198  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    7. S. Kh. Aranson, V. Z. Grines, “Topological classification of flows on closed two-dimensional manifolds”, Russian Math. Surveys, 41:1 (1986), 183–208  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    8. D. V. Anosov, “On the behavior in the Euclidean or Lobachevsky plane of trajectories that cover trajectories of flows on closed surfaces. I”, Math. USSR-Izv., 30:1 (1988), 15–38  mathnet  crossref  mathscinet  zmath
    9. D. V. Anosov, “On the behavior in the Euclidean or Lobachevsky plane of trajectories that cover trajectories of flows on closed surfaces. II”, Math. USSR-Izv., 32:3 (1989), 449–474  mathnet  crossref  mathscinet  zmath
    10. S. Kh. Aranson, E. V. Zhuzhoma, “On the $C^r$-closing lemma on surfaces”, Russian Math. Surveys, 43:5 (1988), 209–210  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    11. D. V. Anosov, “On infinite curves on the Klein bottle”, Math. USSR-Sb., 66:1 (1990), 41–58  mathnet  crossref  mathscinet  zmath  isi
    12. S. Kh. Aranson, E. V. Zhuzhoma, “On the structure of quasiminimal sets of foliations on surfaces”, Russian Acad. Sci. Sb. Math., 82:2 (1995), 397–424  mathnet  crossref  mathscinet  zmath  isi
    13. S. Aranson, E. Zhuzhoma, “Maier's theorems and geodesic laminations of surface flows”, J Dyn Control Syst, 2:4 (1996), 557  crossref  mathscinet  zmath  elib
    14. S. Kh. Aranson, E. V. Zhuzhoma, V. S. Medvedev, “On continuity of geodesic frameworks of flows on surfaces”, Sb. Math., 188:7 (1997), 955–972  mathnet  crossref  crossref  mathscinet  zmath  isi
    15. Mirko Degli Esposti, Gianluigi Del Magno, Marco Lenci, Nonlinearity, 11:4 (1998), 991  crossref  mathscinet  zmath  isi
    16. S. Aranson, E. Zhuzhoma, “Qualitative theory of flows on surfaces (A review)”, Journal of Mathematical Sciences (New York), 90:3 (1998), 2051  crossref  mathscinet  zmath
    17. S. Kh. Aranson, E. V. Zhuzhoma, I. A. Tel'nykh, “Transitive and supertransitive flows on closed nonorientable surfaces”, Math. Notes, 63:4 (1998), 549–552  mathnet  crossref  crossref  mathscinet  zmath  isi
    18. Aranson S., Zhuzhoma E., Medvedev V., “Continuity and Collapse of the Geodesic Framework of Flows on Surfaces”, Dokl. Akad. Nauk, 362:1 (1998), 7–11  mathnet  isi
    19. I. A. Tel'nykh, “Topological classification of supertransitive flows on closed nonorientable surfaces. I: Necessary conditions for topological equivalence”, Math. Notes, 68:3 (2000), 397–404  mathnet  crossref  crossref  mathscinet  zmath  isi
    20. S. Kh. Aranson, E. V. Zhuzhoma, “Properties of the Absolute That Affect Smoothness of Flows on Closed Surfaces”, Math. Notes, 68:6 (2000), 695–703  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    21. S. Aranson, V. Grines, E. Zhuzhoma, “On Anosov–Weil problem”, Topology, 40:3 (2001), 475  crossref
    22. D. V. Anosov, E. V. Zhuzhoma, “Asymptotic Behavior of Covering Curves on the Universal Coverings of Surfaces”, Proc. Steklov Inst. Math., 238 (2002), 1–46  mathnet  mathscinet  zmath
    23. N.G. Markley, M.H. Vanderschoot, “Remote limit points on surfaces”, Journal of Differential Equations, 188:1 (2003), 221  crossref
    24. Aranson S. Grines V. Kaimanovich V., “Classification of Supertransitive 2-Webs on Surfaces”, J. Dyn. Control Syst., 9:4 (2003), 455–468  crossref  isi
    25. S. Kh. Aranson, E. V. Zhuzhoma, “Nonlocal Properties of Analytic Flows on Closed Orientable Surfaces”, Proc. Steklov Inst. Math., 244 (2004), 2–17  mathnet  mathscinet  zmath
    26. D. V. Anosov, E. V. Zhuzhoma, “Nonlocal asymptotic behavior of curves and leaves of laminations on universal coverings”, Proc. Steklov Inst. Math., 249 (2005), 1–221  mathnet  mathscinet  zmath
    27. E. V. Zhuzhoma, “Continuous Dependence of Geodesic Frames in the Hausdorff Metric”, Math. Notes, 77:6 (2005), 862–864  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    28. S. Kh. Aranson, I. A. Gorelikova, E. V. Zhuzhoma, “Closed cross-sections of irrational flows on surfaces”, Sb. Math., 197:2 (2006), 173–192  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    29. N. Markley, M. Vanderschoot, “Recurrent orbits for flows on surfaces”, Journal of Differential Equations, 244:5 (2008), 1210  crossref
    30. Medvedev T.V., “Klassifikatsiya potokov tipa cherri na neorientiruemoi poverkhnosti roda tri”, Vestnik nizhegorodskogo universiteta im. N.I. Lobachevskogo, 2011, no. 2, 139–145  elib
    31. Medvedev T.V., “Klassifikatsiya potokov tipa cherri na neorientiruemoi poverkhnosti roda tri”, Vestnik nizhegorodskogo universiteta im. N.I. Lobachevskogo, 2011, no. 2-1, 139–145  elib
    32. Budnyts'ka N.V., Rybalkina T.V., “Realization of a Closed 1-Form on Closed Oriented Surfaces”, Ukr. Math. J., 64:6 (2012), 844–856  crossref  isi
    33. Alexander Bufetov, “Limit theorems for translation flows”, Ann. Math, 179:2 (2013), 431  crossref
    34. Maksymenko S., Polulyakh E., “Foliations With All Non-Closed Leaves on Non-Compact Surfaces”, Methods Funct. Anal. Topol., 22:3 (2016), 266–282  mathscinet  isi
    35. Grines V., Zhuzhoma E., “Around Anosov-Weil Theory”, Modern Theory of Dynamical Systems: a Tribute to Dmitry Victorovich Anosov, Contemporary Mathematics, 692, eds. Katok A., Pesin Y., Hertz F., Amer Mathematical Soc, 2017, 123–154  crossref  mathscinet  isi  scopus
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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