
Tr. Mat. Inst. Steklova, 2002, Volume 236, Pages 20–26
(Mi tm272)




This article is cited in 2 scientific papers (total in 2 papers)
Flows on Closed Surfaces and Related Geometrical Questions
D. V. Anosov^{} ^{} Steklov Mathematical Institute, Russian Academy of Sciences
Abstract:
When studying flows (continuous oneparameter groups of transformations) on
a surface $M$ (a closed twodimensional manifold, which in our case is
assumed to be different from the sphere and the projective plane), one
naturally faces several geometrical questions related to the behavior of
trajectories lifted to the universal covering plane $\widetilde {M}$ (for
example, the questions of whether the lifted trajectory goes to infinity
and if it has a certain asymptotic direction at infinity). The same
questions can be posed not only for the flow trajectories but also for
leaves of onedimensional foliations and, in general, for
nonselfintersecting (semi)infinite curves. The properties of curves
lifted to $\widetilde M$ that we consider here are such that, if two such
curves $\widetilde L$ and $\widetilde L'$ are situated at a finite
Frechét distance from each other (in this case, we say that the original
curves $L$ and $L'$ are $F$equivalent on $M$), then these properties of
the above curves are identical. Certain (a few) results relate to arbitrary
nonselfintersecting $L$; other results only relate to flow trajectories
under certain additional constraints (that are usually imposed on the set
of equilibrium states). The results of the latter type (which do not hold
for arbitrary nonselfintersecting curves $L$) imply that, in general,
arbitrary $L$ are not $F$equivalent to the trajectories of such flows. In
this relation, nonorientable foliations occupy a kind of intermediate
position.
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English version:
Proceedings of the Steklov Institute of Mathematics, 2002, 236, 12–18
Bibliographic databases:
UDC:
517.91 Received in December 2000
Citation:
D. V. Anosov, “Flows on Closed Surfaces and Related Geometrical Questions”, Differential equations and dynamical systems, Collected papers. Dedicated to the 80th anniversary of academician Evgenii Frolovich Mishchenko, Tr. Mat. Inst. Steklova, 236, Nauka, MAIK «Nauka/Inteperiodika», M., 2002, 20–26; Proc. Steklov Inst. Math., 236 (2002), 12–18
Citation in format AMSBIB
\Bibitem{Ano02}
\by D.~V.~Anosov
\paper Flows on Closed Surfaces and Related Geometrical Questions
\inbook Differential equations and dynamical systems
\bookinfo Collected papers. Dedicated to the 80th anniversary of academician Evgenii Frolovich Mishchenko
\serial Tr. Mat. Inst. Steklova
\yr 2002
\vol 236
\pages 2026
\publ Nauka, MAIK «Nauka/Inteperiodika»
\publaddr M.
\mathnet{http://mi.mathnet.ru/tm272}
\mathscinet{http://www.ams.org/mathscinetgetitem?mr=1931002}
\zmath{https://zbmath.org/?q=an:1020.37022}
\transl
\jour Proc. Steklov Inst. Math.
\yr 2002
\vol 236
\pages 1218
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This publication is cited in the following articles:

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

Grines V. Zhuzhoma E., “Around AnosovWeil Theory”, Modern Theory of Dynamical Systems: a Tribute to Dmitry Victorovich Anosov, Contemporary Mathematics, 692, ed. Katok A. Pesin Y. Hertz F., Amer Mathematical Soc, 2017, 123–154

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