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Izv. Akad. Nauk SSSR Ser. Mat., 1982, Volume 46, Issue 4, Pages 675–709 (Mi izv1639)  

This article is cited in 12 scientific papers (total in 14 papers)

On generic properties of closed geodesics

D. V. Anosov


Abstract: A complete proof is given of the theorem asserting that bumpy metrics are generic. This result was announced by Abraham (Global Analysis (Proc. Sympos. Pure Math., vol 14), Amer. Math. Soc., Providence, R. I., 1970, pp. 1–3.). Related results are used to rigourously carry out Poincaré's outline of a “bifurcation-theoretic” proof of the existence of closed geodesics without self-intersections for any Riemannian metric of positive curvature on the two-dimensional sphere $S^2$. To do this, it is essential that the lengths of all non-self-intersecting closed geodesics for the metrics on $S^2$, considered in the course of the proof, be uniformly bounded from above. Examples are given of $C^\infty$ metrics on $S^2$ (where the sign of the curvature alternates) for which there exist arbitrarily long closed geodesics without self-intersections.
Bibliography: 27 titles.

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English version:
Mathematics of the USSR-Izvestiya, 1983, 21:1, 1–29

Bibliographic databases:

UDC: 513.78+517.9
MSC: Primary 53C22; Secondary 58F17, 58F22
Received: 15.03.1982

Citation: D. V. Anosov, “On generic properties of closed geodesics”, Izv. Akad. Nauk SSSR Ser. Mat., 46:4 (1982), 675–709; Math. USSR-Izv., 21:1 (1983), 1–29

Citation in format AMSBIB
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\by D.~V.~Anosov
\paper On generic properties of closed geodesics
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1982
\vol 46
\issue 4
\pages 675--709
\mathnet{http://mi.mathnet.ru/izv1639}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=670163}
\zmath{https://zbmath.org/?q=an:0554.58043|0512.58014}
\transl
\jour Math. USSR-Izv.
\yr 1983
\vol 21
\issue 1
\pages 1--29
\crossref{https://doi.org/10.1070/IM1983v021n01ABEH001637}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. V. V. Kozlov, “Calculus of variations in the large and classical mechanics”, Russian Math. Surveys, 40:2 (1985), 37–71  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    2. A. T. Fomenko, “The topology of surfaces of constant energy in integrable Hamiltonian systems, and obstructions to integrability”, Math. USSR-Izv., 29:3 (1987), 629–658  mathnet  crossref  mathscinet  zmath
    3. A. T. Fomenko, H. Zieschang, “On typical topological properties of integrable Hamiltonian systems”, Math. USSR-Izv., 32:2 (1989), 385–412  mathnet  crossref  mathscinet  zmath
    4. A. T. Fomenko, “The symplectic topology of completely integrable Hamiltonian systems”, Russian Math. Surveys, 44:1 (1989), 181–219  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    5. I. A. Taimanov, “Closed extremals on two-dimensional manifolds”, Russian Math. Surveys, 47:2 (1992), 163–211  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    6. 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  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    7. Paternain G.P., Petean J., “On the growth rate of contractible closed geodesics on reducible manifolds”, Geometry and Dynamics, Contemporary Mathematics Series, 389, 2005, 191–196  isi
    8. Ivan Kupka, Mauricio Peixoto, Charles Pugh, “Focal stability of Riemann metrics”, crll, 2006:593 (2006), 31  crossref  mathscinet  isi
    9. I. A. Taimanov, “The type numbers of closed geodesics”, Reg Chaot Dyn, 15:1 (2010), 84  crossref  isi  elib
    10. Gonzalo Contreras, “Geodesic flows with positive topological entropy, twist maps and hyperbolicity”, Ann of Math, 172:2 (2010), 761  crossref
    11. Takeshi Isobe, “On the existence of nonlinear Dirac-geodesics on compact manifolds”, Calc. Var, 2011  crossref
    12. Mário Bessa, M.J.oana Torres, “The <mml:math altimg="si1.gif" overflow="scroll" xmlns:xocs="http://www.elsevier.com/xml/xocs/dtd" xmlns:xs="http://www.w3.org/2001/XMLSchema" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns="http://www.elsevier.com/xml/ja/dtd" xmlns:ja="http://www.elsevier.com/xml/ja/dtd" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:tb="http://www.elsevier.com/xml/common/table/dtd" xmlns:sb="http://www.elsevier.com/xml/common/struct-bib/dtd" xmlns:ce="http://www.elsevier.com/xml/common/dtd" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:cals="http://www.elsevier.com/xml/common/cals/dtd" xmlns:sa="http://www.elsevier.com/xml/common/struct-aff/dtd"><mml:msup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msup></mml:math> general density theorem for geodesic flows”, Comptes Rendus Mathematique, 2015  crossref
    13. S. M. Aseev, V. M. Buchstaber, R. I. Grigorchuk, V. Z. Grines, B. M. Gurevich, A. A. Davydov, A. Yu. Zhirov, E. V. Zhuzhoma, M. I. Zelikin, A. B. Katok, A. V. Klimenko, V. V. Kozlov, V. P. Leksin, M. I. Monastyrskii, A. I. Neishtadt, S. P. Novikov, E. A. Sataev, Ya. G. Sinai, A. M. Stepin, “Dmitrii Viktorovich Anosov (obituary)”, Russian Math. Surveys, 70:2 (2015), 369–381  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    14. I. V. Sypchenko, D. S. Timonina, “Closed geodesics on piecewise smooth surfaces of revolution with constant curvature”, Sb. Math., 206:5 (2015), 738–769  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
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