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 Izv. RAN. Ser. Mat., 1995, Volume 59, Issue 2, Pages 63–96 (Mi izv12)

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

D. V. Anosov

Abstract: This paper is related to the previous papers [1] and [2]. We fill a gap in the proof in [1] of the following alternative: under assumptions mentioned there, a semi-trajectory $\widetilde L$ of the covering flow on the universal covering plane is either bounded or tends to infinity with an asymptotic direction. For the torus, we prove under the same assumptions that in the second case the deviation of $\widetilde L$ from the line corresponding to this direction is bounded. We prove that for every (semi-)infinite non-self-intersecting $L$ on a closed surface and every $r>0$ there is a $C^\infty$-flow with an invariant measure having a specified $C^\infty$-smooth everywhere-positive density such that some positive semi-trajectory of the flow approximates $L$ up to $r$. (In [2] an analogous approximation assertion was proved, with no mention of an invariant measure.)

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English version:
Izvestiya: Mathematics, 1995, 59:2, 287–320

Bibliographic databases:

MSC: 58F25

Citation: D. V. Anosov, “On the behaviour in the Euclidean or Lobachevsky plane of trajectories that cover trajectories of flows on closed surfaces. III”, Izv. RAN. Ser. Mat., 59:2 (1995), 63–96; Izv. Math., 59:2 (1995), 287–320

Citation in format AMSBIB
\Bibitem{Ano95}
\by D.~V.~Anosov
\paper On the behaviour in the Euclidean or Lobachevsky plane of trajectories that cover trajectories of flows on closed surfaces.~III
\jour Izv. RAN. Ser. Mat.
\yr 1995
\vol 59
\issue 2
\pages 63--96
\mathnet{http://mi.mathnet.ru/izv12}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1337159}
\zmath{https://zbmath.org/?q=an:0902.58031}
\transl
\jour Izv. Math.
\yr 1995
\vol 59
\issue 2
\pages 287--320
\crossref{https://doi.org/10.1070/IM1995v059n02ABEH000012}

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This publication is cited in the following articles:
1. 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
2. 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
3. D. V. Anosov, “Flows on Closed Surfaces and Related Geometrical Questions”, Proc. Steklov Inst. Math., 236 (2002), 12–18
4. 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
5. S. Kh. Aranson, E. V. Zhuzhoma, “On asymptotic directions of semitrajectories of analytic flows on surfaces”, Russian Math. Surveys, 57:6 (2002), 1207–1209
6. S. Kh. Aranson, E. V. Zhuzhoma, “Nonlocal Properties of Analytic Flows on Closed Orientable Surfaces”, Proc. Steklov Inst. Math., 244 (2004), 2–17
7. 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
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