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Izv. Akad. Nauk SSSR Ser. Mat., 1987, Volume 51, Issue 1, Pages 16–43 (Mi izv1261)  

This article is cited in 20 scientific papers (total in 22 papers)

On the behavior in the Euclidean or Lobachevsky plane of trajectories that cover trajectories of flows on closed surfaces. I

D. V. Anosov


Abstract: Consider a flow on a surface $M$ of nonpositive Euler characteristic whose set of equilibrium points can be deformed, in $M$, to a point (this, for example, is the case if there are only finitely many equilibrium points). For such a flow, it is proved that the semitrajectory of the covering flow on the universal cover (the Euclidean or Lobachevsky plane) of $M$ is either bounded or tends to infinity in a definite direction. For analytic flows (but not for $C^\infty$-flows), this conclusion holds without any conditions on the equilibrium points.
Bibliography: 21 titles.

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English version:
Mathematics of the USSR-Izvestiya, 1988, 30:1, 15–38

Bibliographic databases:

Document Type: Article
UDC: 517.91
MSC: Primary 58F25; Secondary 34C35, 34C40
Received: 16.02.1986

Citation: D. V. Anosov, “On the behavior in the Euclidean or Lobachevsky plane of trajectories that cover trajectories of flows on closed surfaces. I”, Izv. Akad. Nauk SSSR Ser. Mat., 51:1 (1987), 16–43; Math. USSR-Izv., 30:1 (1988), 15–38

Citation in format AMSBIB
\Bibitem{Ano87}
\by D.~V.~Anosov
\paper On the behavior in the Euclidean or Lobachevsky plane of trajectories that cover trajectories of flows on closed surfaces.~I
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1987
\vol 51
\issue 1
\pages 16--43
\mathnet{http://mi.mathnet.ru/izv1261}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=887599}
\zmath{https://zbmath.org/?q=an:0637.58025|0621.58032}
\transl
\jour Math. USSR-Izv.
\yr 1988
\vol 30
\issue 1
\pages 15--38
\crossref{https://doi.org/10.1070/IM1988v030n01ABEH000990}


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    This publication is cited in the following articles:
    1. 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
    2. D. V. Anosov, “On infinite curves on the Klein bottle”, Math. USSR-Sb., 66:1 (1990), 41–58  mathnet  crossref  mathscinet  zmath  isi
    3. S. Kh. Aranson, E. V. Zhuzhoma, “Trajectories covering flows for branched coverings of the sphere and projective plane”, Math. Notes, 53:5 (1993), 463–468  mathnet  crossref  mathscinet  zmath  isi  elib
    4. D. V. Anosov, “Flows on closed surfaces and behavior of trajectories lifted to the universal covering plane”, J Dyn Control Syst, 1:1 (1995), 125  crossref  mathscinet  zmath  elib
    5. S. Kh. Aranson, V. Z. Grines, E. V. Zhuzhoma, “On the geometry and topology of flows and foliations on surfaces and the Anosov problem”, Sb. Math., 186:8 (1995), 1107–1146  mathnet  crossref  mathscinet  zmath  isi
    6. D. V. Anosov, “On the behaviour in the Euclidean or Lobachevsky plane of trajectories that cover trajectories of flows on closed surfaces. III”, Izv. Math., 59:2 (1995), 287–320  mathnet  crossref  mathscinet  zmath  isi
    7. 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
    8. V. I. Arnol'd, A. A. Bolibrukh, R. V. Gamkrelidze, V. P. Maslov, E. F. Mishchenko, S. P. Novikov, Yu. S. Osipov, Ya. G. Sinai, A. M. Stepin, L. D. Faddeev, “Dmitrii Viktorovich Anosov (on his 60th birthday)”, Russian Math. Surveys, 52:2 (1997), 437–445  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    9. A. A. Glutsyuk, “Limit sets at infinity for liftings of non-self-intersecting curves on the torus to the plane”, Math. Notes, 64:5 (1998), 579–589  mathnet  crossref  crossref  mathscinet  zmath  isi
    10. D. V. Anosov, “On the Lifts to the Plane of Semileaves of Foliations on the Torus with a Finite Number of Singularities”, Proc. Steklov Inst. Math., 224 (1999), 20–45  mathnet  mathscinet  zmath
    11. 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
    12. João Lopes Dias, Nonlinearity, 15:3 (2002), 665  crossref  mathscinet  zmath  isi
    13. D. V. Anosov, “Flows on Closed Surfaces and Related Geometrical Questions”, Proc. Steklov Inst. Math., 236 (2002), 12–18  mathnet  mathscinet  zmath
    14. 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
    15. S. Kh. Aranson, E. V. Zhuzhoma, “On asymptotic directions of semitrajectories of analytic flows on surfaces”, Russian Math. Surveys, 57:6 (2002), 1207–1209  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    16. N.G. Markley, M.H. Vanderschoot, “Remote limit points on surfaces”, Journal of Differential Equations, 188:1 (2003), 221  crossref
    17. 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
    18. 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
    19. 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
    20. 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
    21. N. Markley, M. Vanderschoot, “Recurrent orbits for flows on surfaces”, Journal of Differential Equations, 244:5 (2008), 1210  crossref
    22. S. M. Aseev, V. M. Buchstaber, R. I. Grigorchuk, V. Z. Grines, B. M. Gurevich, A. A. Davydov, A. Yu. Zhirov, E. V. Zhuzhoma, M. I. Zelikin, A. B. Katok, A. V. Klimenko, V. V. Kozlov, V. P. Leksin, M. I. Monastyrskii, A. I. Neishtadt, S. P. Novikov, E. A. Sataev, Ya. G. Sinai, A. M. Stepin, “Dmitrii Viktorovich Anosov (obituary)”, Russian Math. Surveys, 70:2 (2015), 369–381  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
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