Abstract:
Foliations on compact surfaces are considered in this paper. The structure of a quasiminimal set is studied, and criteria for the recurrence of a nonclosed leaf are proved. The concept of an amply situated quasiminimal set is introduced, and the nonexistence of such sets on some orientable and nonorientable surfaces is proved. A sharp estimate of the number of quasiminimal sets of foliations on compact surfaces is given. These results are applied to an estimate of the number of one-dimensional basic sets of AA-diffeomorphisms of surfaces.
Citation:
S. Kh. Aranson, E. V. Zhuzhoma, “On the structure of quasiminimal sets of foliations on surfaces”, Russian Acad. Sci. Sb. Math., 82:2 (1995), 397–424
\Bibitem{AraZhu94}
\by S.~Kh.~Aranson, E.~V.~Zhuzhoma
\paper On the structure of quasiminimal sets of foliations on surfaces
\jour Russian Acad. Sci. Sb. Math.
\yr 1995
\vol 82
\issue 2
\pages 397--424
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\crossref{https://doi.org/10.1070/SM1995v082n02ABEH003572}
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https://www.mathnet.ru/eng/sm917
https://doi.org/10.1070/SM1995v082n02ABEH003572
https://www.mathnet.ru/eng/sm/v185/i8/p31
This publication is cited in the following 15 articles:
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
Irina Gelbukh, “Structure of a Morse form foliation on a closed surface in terms of genus”, Differential Geometry and its Applications, 2011
A. López, “Foliation Admitting Recurrent Leaves of Infinite Depth on Compact Two-Manifolds”, J Dyn Control Syst, 13:2 (2007), 255
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., “On Structurally Stable Diffeomorphisms with Codimension One Expanding Attractors”, Trans. Am. Math. Soc., 357:2 (2005), 617–667
Lopez A., “A Structure Theorem for Foliations on Non-Compact 2-Manifolds”, Ergod. Theory Dyn. Syst., 25:3 (2005), 893–912
Gutierrez C., Hector G., Lopez A., “Interval Exchange Transformations and Foliations on Infinite Genus 2-Manifolds”, Ergod. Theory Dyn. Syst., 24:4 (2004), 1097–1108
Aranson S., Grines V., Kaimanovich V., “Classification of Supertransitive 2-Webs on Surfaces”, J. Dyn. Control Syst., 9:4 (2003), 455–468
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
V. Z. Grines, E. V. Zhuzhoma, “Structurally stable diffeomorphisms with basis sets of codimension one”, Izv. Math., 66:2 (2002), 223–284
S. Aranson, V. Grines, E. Zhuzhoma, “On Anosov–Weil problem”, Topology, 40:3 (2001), 475
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
S. Kh. Aranson, R. V. Plykin, A. Yu. Zhirov, E. V. Zhuzhoma, “Exact upper bounds for the number of one-dimensional basic sets of surfaceA-diffeomorphisms”, J Dyn Control Syst, 3:1 (1997), 1
Nikolaev I., “The Poincaré-Bendixson Theorem and Arational Foliations on the Sphere”, Ann. Inst. Fourier, 46:4 (1996), 1159–&
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
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