Advertisement

Mathematical Notes

, Volume 61, Issue 4, pp 473–479 | Cite as

Recurrence of the integral of an odd conditionally periodic function

  • S. V. Konyagin
Article

Abstract

We prove that the integral of a sufficiently smooth odd conditionally periodic function with zero mean and incommensurable frequencies recurs. Furthermore, we obtain the lower and upper bounds for smoothness guaranteeing the recurrence of the integral.

Key words

recurrence of an integral conditionally periodic function 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    H. Poincaré,On curves defined by differential equations [Russian translation], Gostekhizdat, Moscow-Leningrad (1947).Google Scholar
  2. 2.
    E. A. Sidorov, “On conditions for uniform stability (in the sense of Poisson) of cylindrical systems,”Uspekhi Mat. Nauk [Russian Math. Surveys],34, No. 6, 184–188 (1979).zbMATHMathSciNetGoogle Scholar
  3. 3.
    V. V. Kozlov,Methods of Qualitative Analysis in Dynamics of Solids [in Russian], Izd. Moskov. Univ., Moscow (1980).Google Scholar
  4. 4.
    N. G. Moshchevitin, “Recurrence of the integral of a smooth three-frequency conditionally periodic function,”Mat. Zametki [Math. Notes],58, No. 5, 723–735 (1985).MathSciNetGoogle Scholar
  5. 5.
    N. G. Moshchevitin, “Nonrecurrence of the integral of a conditionally periodic function,”Mat. Zametki [Math. Notes],49, No. 6, 138–140 (1991).zbMATHMathSciNetGoogle Scholar
  6. 6.
    V. G. Sprindzhuk, “Asymptotic behavior of the integrals of quasiperiodic functions,”Differentsial'nye Uravneniya [Differential Equations],3, No. 6, 862–868 (1967).zbMATHGoogle Scholar
  7. 7.
    V. G. Sprindzhuk, “Quasiperiodic functions with unbounded indefinite integral,” Dokl. Akad. Nauk BSSR,12, No. 1, 5–8 (1968).zbMATHMathSciNetGoogle Scholar
  8. 8.
    L. G. Peck, “On uniform distribution of algebraic numbers,”Proc. Amer. Math. Soc.,4, No. 1, 440–443 (1953).zbMATHMathSciNetGoogle Scholar
  9. 9.
    N. G. Moshchevitin, “The final properties of integrals of conditionally periodic functions related to the problem of small denominators,”Vestnik Moskov. Univ. Ser. I Mat. Mekh. [Moscow Univ. Math. Bull.] No. 5, 94–96 (1988).zbMATHMathSciNetGoogle Scholar
  10. 10.
    N. G. Moshchevitin, “The uniform distribution of the fractional parts of the systems of linear functions,”Vestnik Moskov. Univ. Ser. I Mat. Mekh. [Moscow Univ. Math. Bull.] No. 4, 26–31 (1990).zbMATHMathSciNetGoogle Scholar
  11. 11.
    N. G. Moshchevitin, “The behavior of the integral of a conditionally periodic function,”Mat. Zametki [Math. Notes],50, No. 3, 97–106 (1991).zbMATHMathSciNetGoogle Scholar
  12. 12.
    N. G. Moshchevitin, “Distribution of values of linear functions and asymptotic behavior of trajectories of some dynamical systems,”Mat. Zametki [Math. Notes],58, No. 3, 394–410 (1995).zbMATHMathSciNetGoogle Scholar

Copyright information

© Plenum Publishing Corporation 1997

Authors and Affiliations

  • S. V. Konyagin
    • 1
  1. 1.M. V. Lomonosov Moscow State UniversityMoscowUSSR

Personalised recommendations