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 Mat. Sb., 1989, Volume 180, Number 1, Pages 39–56 (Mi msb1597)

On infinite curves on the Klein bottle

D. V. Anosov

Abstract: The author investigates continuous nonselfintersecting (semi-) infinite curves $L=ż(t);t\geqslant0\}$ on the Klein bottle $\mathbf R^2/\Gamma$, where the group $\Gamma$ of covering transformations is generated by translations through elements of the integral lattice together with the transformation $(x,y)\mapsto(x+\frac12,-y)$. It is proved that if $\widetilde L=\{\widetilde z(t)\}\subset\mathbf R^2$ is a curve which covers $L$ and goes to infinity, then $\widetilde L$ has a horizontal or vertical asymptotic direction $\widetilde l$ at infinity; that is, a ray starting at a fixed point of $\mathbf R^2$ and passing through $\widetilde z(t)$ has a horizontal or vertical limit as $t\to\infty$. In the first case (when $\widetilde l$ is horizontal) the divergence of $\widetilde L$ from $\widetilde l$ is bounded, but in the second case it can be unbounded on one side (but not on both). In passing, a simplified description is given of an example (published earlier in Trudy Mat. Inst. Steklov. 185 (1988), 30–35) demonstrating the existence of the analogous phenomenon of unbounded divergence for the torus.
Bibliography: 8 titles.

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English version:
Mathematics of the USSR-Sbornik, 1990, 66:1, 41–58

Bibliographic databases:

UDC: 517.91
MSC: Primary 58F25; Secondary 34C35, 34C40

Citation: D. V. Anosov, “On infinite curves on the Klein bottle”, Mat. Sb., 180:1 (1989), 39–56; Math. USSR-Sb., 66:1 (1990), 41–58

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. 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
2. 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
3. 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
4. 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
5. 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
6. 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
7. S. Aranson, V. Grines, E. Zhuzhoma, “On Anosov–Weil problem”, Topology, 40:3 (2001), 475
8. D. V. Anosov, “Flows on Closed Surfaces and Related Geometrical Questions”, Proc. Steklov Inst. Math., 236 (2002), 12–18
9. 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
10. 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
11. 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
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