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 Mat. Zametki, 2000, Volume 68, Issue 6, Pages 819–829 (Mi mz1004)

Properties of the Absolute That Affect Smoothness of Flows on Closed Surfaces

S. Kh. Aranson, E. V. Zhuzhoma

Nizhny Novgorod State Technical University

Abstract: Let $M^2_g$ be a closed orientable surface of genus $g\ge2$, endowed with the structure of a Riemann manifold of constant negative curvature. For the universal covering $\Delta$, there is the notion of absolute, each of whose points determines an asymptotic direction of a bundle of parallel equidirected geodesics. In the paper it is proved that there is a set $U_g$ on the absolute having the cardinality of the continuum and such that if an arbitrary flow on $M^2_g$ has a semitrajectory whose covering has asymptotic direction defined by a point from $U_g$, then this flow is not analytical and has infinitely many stationary points.

DOI: https://doi.org/10.4213/mzm1004

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English version:
Mathematical Notes, 2000, 68:6, 695–703

Bibliographic databases:

UDC: 517.917+513.9

Citation: S. Kh. Aranson, E. V. Zhuzhoma, “Properties of the Absolute That Affect Smoothness of Flows on Closed Surfaces”, Mat. Zametki, 68:6 (2000), 819–829; Math. Notes, 68:6 (2000), 695–703

Citation in format AMSBIB
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• https://doi.org/10.4213/mzm1004
• http://mi.mathnet.ru/eng/mz/v68/i6/p819

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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. Aranson, SK, “The influence of the absolute on the local and smooth properties of foliations and homeomorphisms with invariant foliations on closed surfaces”, Doklady Mathematics, 64:1 (2001), 25
2. S. Kh. Aranson, E. V. Zhuzhoma, “On asymptotic directions of semitrajectories of analytic flows on surfaces”, Russian Math. Surveys, 57:6 (2002), 1207–1209
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, “Nonlocal Properties of Analytic Flows on Closed Orientable Surfaces”, Proc. Steklov Inst. Math., 244 (2004), 2–17
6. 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
7. Grines V. Zhuzhoma E., “Around Anosov-Weil 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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