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Izv. Akad. Nauk SSSR Ser. Mat., 1973, Volume 37, Issue 3, Pages 539–576 (Mi izv2279)  

This article is cited in 20 scientific papers (total in 20 papers)

Ergodic perturbations of degenerate integrable Hamiltonian systems

A. B. Katok


Abstract: Hamiltonian systems arbitrarily close in the $C^r$ topology $(r=1,2,…)$ to a given integrable degenerate Hamiltonian system of class $C^\infty$ which generate an ergodic flow on each manifold of constant energy are constructed. Applications: Small perturbations of a system generated by independent oscillators and Finsler metrics close to standard Riemannian metrics on symmetric spaces of rank 1.

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English version:
Mathematics of the USSR-Izvestiya, 1973, 7:3, 535–571

Bibliographic databases:

UDC: 517.9+513.78
MSC: Primary 58F05, 28A65; Secondary 53C60
Received: 02.10.1972

Citation: A. B. Katok, “Ergodic perturbations of degenerate integrable Hamiltonian systems”, Izv. Akad. Nauk SSSR Ser. Mat., 37:3 (1973), 539–576; Math. USSR-Izv., 7:3 (1973), 535–571

Citation in format AMSBIB
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\by A.~B.~Katok
\paper Ergodic perturbations of degenerate integrable Hamiltonian systems
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1973
\vol 37
\issue 3
\pages 539--576
\mathnet{http://mi.mathnet.ru/izv2279}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=331425}
\zmath{https://zbmath.org/?q=an:0316.58010}
\transl
\jour Math. USSR-Izv.
\yr 1973
\vol 7
\issue 3
\pages 535--571
\crossref{https://doi.org/10.1070/IM1973v007n03ABEH001958}


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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. O Penrose, Rep Prog Phys, 42:12 (1979), 1937  crossref  adsnasa  isi
    2. Jürgen Pöschel, “Integrability of Hamiltonian systems on Cantor sets”, Comm Pure Appl Math, 35:5 (1982), 653  crossref  mathscinet
    3. K. Burns, A. Katok, W. Ballman, M. Brin, P. Eberlein, R. Osserman, “Manifolds with non-positive curvature”, Ergod Th Dynam Sys, 5:2 (1985)  crossref  mathscinet  zmath
    4. I. A. Taimanov, “Closed extremals on two-dimensional manifolds”, Russian Math. Surveys, 47:2 (1992), 163–211  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    5. M. B. Sevryuk, “The classical KAM theory at the dawn of the twenty-first century”, Mosc. Math. J., 3:3 (2003), 1113–1144  mathnet  mathscinet  zmath
    6. D. V. Anosov, “Spectral Multiplicity in Ergodic Theory”, Proc. Steklov Inst. Math., 290, suppl. 1 (2015), 1–44  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    7. Per Christian Moan, “On the KAM and Nekhoroshev theorems for symplectic integrators and implications for error growth”, Nonlinearity, 17:1 (2004), 67  crossref  mathscinet  zmath  isi  elib
    8. Adam Harris, Gabriel P. Paternain, “Dynamically convex Finsler metrics and J-holomorphic embedding of asymptotic cylinders”, Ann Global Anal Geom, 34:2 (2008), 115  crossref  mathscinet  zmath  isi
    9. G. W. Gibbons, C. A. R. Herdeiro, C. M. Warnick, M. C. Werner, “Stationary metrics and optical Zermelo-Randers-Finsler geometry”, Phys Rev D, 79:4 (2009), 044022  crossref  adsnasa  isi
    10. Duan Huagui, Long Yiming, “Morse concavity for closed geodesics”, Acta Mathematica Scientia, 29:3 (2009), 731  crossref
    11. I. A. Taimanov, “The type numbers of closed geodesics”, Reg Chaot Dyn, 15:1 (2010), 84  crossref  isi  elib
    12. Wei Wang, “Closed geodesics on Finsler spheres”, Calc. Var, 45:1-2 (2011), 253  crossref
    13. Wei Wang, “On the average indices of closed geodesics on positively curved Finsler spheres”, Math. Ann, 355:3 (2013), 1049  crossref
    14. J.P.hilipp Schröder, “Global minimizers for Tonelli Lagrangians on the 2-torus”, J. Topol. Anal, 2014, 1  crossref
    15. S. Yu. Dobrokhotov, D. S. Minenkov, M. Rouleux, “The Maupertuis–Jacobi Principle for Hamiltonians of the Form $F(x,|p|)$ in Two-Dimensional Stationary Semiclassical Problems”, Math. Notes, 97:1 (2015), 42–49  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    16. J.P.HILIPP SCHRÖDER, “Invariant tori and topological entropy in Tonelli Lagrangian systems on the 2-torus”, Ergod. Th. Dynam. Sys, 2015, 1  crossref
    17. Huagui Duan, Yiming Long, Yuming Xiao, “Two closed geodesics on
      $${\mathbb {R}}P^{2n+1}$$
      R P 2 n + 1 with a bumpy Finsler metric”, Calc. Var, 2015  crossref
    18. J.P.hilipp Schröder, “Ergodicity and topological entropy of geodesic flows on surfaces”, JMD, 9:01 (2015), 147  crossref
    19. I. A. Taimanov, “The spaces of non-contractible closed curves in compact space forms”, Sb. Math., 207:10 (2016), 1458–1470  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    20. Yu. A. Kordyukov, I. A. Taimanov, “Formula sleda dlya magnitnogo laplasiana”, UMN, 74:2(446) (2019), 149–186  mathnet  crossref  elib
  • Известия Академии наук СССР. Серия математическая Izvestiya: Mathematics
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