Mathematical Notes

, Volume 93, Issue 1–2, pp 83–89 | Cite as

On Bohl’s argument theorem

  • V. V. Kozlov


The classical Bohl argument theorem of a conditionally periodic function is generalized. Conditionally periodic motions on a torus are replaced by the solutions of a nonlinear system of differential equations with invariant measure. Cases in which this system is assumed ergodic or strictly ergodic are considered.


Bohl’s argument theorem conditionally periodic motion on the n-dimensional torus (strictly) ergodic system of differential equations uniformly distributed function Birkhoff-Khinchine ergodic theorem 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    P. Bohl, “Über eine Differentialgleichung der Störungstheorie,” J. Reine Angew. Math. 131, 268–321 (1906).zbMATHGoogle Scholar
  2. 2.
    B.M. Levitan, Almost Periodic Functions (Gostekhizdat, Moscow, 1953) [in Russian].zbMATHGoogle Scholar
  3. 3.
    J. Moser, “On the volume elements on a manifold,” Trans. Amer. Math. Soc. 120(2), 286–294 (1965).MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    N. Kryloff and N. Bogoliouboff, “La theorie génerale de la mesure dans son application à l’etude des systèmes dynamiques de la mécanique non linéare,” Ann. of Math. (2) 38(1), 65–113 (1937).MathSciNetCrossRefGoogle Scholar
  5. 5.
    J. Moser, “A rapidly convergent iterationmethod and nonlinear differential equations. II,” Ann. Scuola Norm. Sup. Pisa (3) 20(3), 499–535 (1966).MathSciNetGoogle Scholar
  6. 6.
    A. N. Kolmogorov, “On dynamical systems with an integral invariant on the torus,” Dokl. Akad. Nauk SSSR 93(5), 763–766 (1953).MathSciNetzbMATHGoogle Scholar
  7. 7.
    A. Poincaré, Probability Theory (RKhD, Izhevsk, 1999) [in Russian].zbMATHGoogle Scholar
  8. 8.
    V. V. Kozlov, “Dynamical systems on the torus with multivalued integrals,” in Trudy Mat. Inst. Steklov, Vol. 256: Dynamical Systems and Optimization, Collection of papers dedicated to the 70th anniversary of Academician D. V. Anosov (Nauka, Moscow, 2007), pp. 201–218 [Proc. Steklov Inst. Math 256, 188–205 (2007)].Google Scholar
  9. 9.
    G. Halász, “Remarks on the remainder in Birkhoff’s ergodic theorem,” Acta Math. Acad. Sci. Hungar. 28(3–4), 389–395 (1976).MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    V. V. Kozlov, “Weighted means, strict ergodicity, and uniform distributions,” Mat. Zametki 78(3), 358–367 (2005) [Math. Notes 78 (3), 329–337 (2005)].MathSciNetCrossRefGoogle Scholar
  11. 11.
    S. A. Chaplygin, “On the theory of motion of nonholonomic systems: A reducing multiplier theorem,” Mat. Sb. 28(2), 303–314 (1912).Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2013

Authors and Affiliations

  1. 1.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia

Personalised recommendations