Limit Distributions for Conflict Dynamical System with Point Spectra

  • V. D. Koshmanenko
  • V. O. Voloshyna

We construct a model of conflict dynamical system whose limit states are associated with singular distributions. It is proved that a criterion for the appearance of point spectrum in the limit distribution is the strategy with fixed priority. In all other cases, the limit distributions are pure singular continuous.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    S. Albeverio, V. Koshmanenko, M. Pratsiovytyi, et al., “Spectral properties of image measures under infinite conflict interactions,” Positivity, 10, 39–49 (2006).MathSciNetzbMATHGoogle Scholar
  2. 2.
    V. Koshmanenko and N. Kharchenko, “Spectral properties of image measures after conflict interactions,” Theory Stochast. Process., 10(26), No. 3-4, 73–81 (2004).Google Scholar
  3. 3.
    V. D. Koshmanenko and N. V. Kharchenko, “Invariant points of a dynamical system of conflict in the space of piecewise-uniformly distributed measures,” Ukr. Mat. Zh., 56, No. 7, 927–938 (2004); English translation : Ukr. Math. J., 56, No. 7, 1102–1116 (2004).Google Scholar
  4. 4.
    S. Albeverio, V. Koshmanenko, and G. Torbin, “Fine structure of the singular continuous spectrum,” Methods Funct. Anal. Top., 9, No. 2, 101–119 (2003).MathSciNetzbMATHGoogle Scholar
  5. 5.
    S. Albeverio, V. Koshmanenko, M. Pratsiovytyi, et al., “On fine structure of singularly continuous probability measures and random variables with independent Ǭ-symbols,” Methods Funct. Anal. Top., 17, No. 2, 97–111 (2011).MathSciNetzbMATHGoogle Scholar
  6. 6.
    H. M. Torbin, “Multifractal analysis of singularly continuous probability measures,” Ukr. Mat. Zh., 57, No. 5, 706–720 (2005); English translation : Ukr. Math. J., 57, No. 5, 837–857 (2005).Google Scholar
  7. 7.
    G. M. Torbin, “Fractal properties of the distributions of random variables with independent Q-symbols,” Trans. Nat. Pedagog. Univ. (Phys.-Math. Sci.), 3, 241–252 (2002).Google Scholar
  8. 8.
    V. D. Koshmanenko, “Full measure of a set of singular continuous measures,” Ukr. Mat. Zh., 61, No. 1, 83–91 (2009); English translation: Ukr. Math. J., 61, No. 1, 99–111 (2009).Google Scholar
  9. 9.
    M. V. Bondarchuk, V. D. Koshmanenko, and N. V. Kharchenko, “Properties of the limit states of a dynamical conflict system,” Nelin. Kolyv., 7, No. 4, 446–461 (2004); English translation : Nonlin. Oscillat., 7, No. 4, 432–447 (2004).Google Scholar
  10. 10.
    V. D. Koshmanenko, “Theorem on conflict for a pair of stochastic vectors,” Ukr. Mat. Zh., 55, No. 4, 555–560 (2003); English translation : Ukr. Math. J., 55, No. 4, 671–678 (2003).Google Scholar
  11. 11.
    V. Koshmanenko, “The theorem of conflict for probability measures,” Math. Meth. Oper. Res., 59, No. 2, 303–313 (2004).MathSciNetzbMATHGoogle Scholar
  12. 12.
    S. Albeverio, M. Bodnarchyk, and V. Koshmanenko, “Dynamics of discrete conflict interactions between nonannihilating opponents,” Meth. Funct. Anal. Top., 11, No. 4, 309–319 (2005).zbMATHGoogle Scholar
  13. 13.
    V. Koshmanenko, Spectral Theory for Conflict Dynamical Systems [in Ukrainian], Naukova Dumka, Kyiv (2016).Google Scholar
  14. 14.
    T. Zamfirescu, “Most monotone functions are singular,” Amer. Math. Monthly, 88, 47–49 (1981).MathSciNetzbMATHGoogle Scholar
  15. 15.
    R. del Rio, S. Jitomirskaya, N. Makarov, et al., “Operators with singular continuous spectrum are generic,” Bull. Amer. Math. Soc., 31, 208–212 (1994).MathSciNetzbMATHGoogle Scholar
  16. 16.
    B. Simon, “Operators with singular continuous spectrum: I. General operators,” Ann. Math., 141, 131–145 (1995).MathSciNetzbMATHGoogle Scholar
  17. 17.
    B. Mandelbrot, Fractals: Form, Chance, and Dimension, Freeman & Co., San Francisco (1977).zbMATHGoogle Scholar
  18. 18.
    M. F. Barnsley, Fractals Everywhere, Academic Press, Boston (1988).zbMATHGoogle Scholar
  19. 19.
    M. F. Barnsley and S. Demko, “Iterated functional system and the global construction of fractals,” Proc. Roy. Soc. London A, 399, 243–275 (1985).MathSciNetzbMATHGoogle Scholar
  20. 20.
    K. J. Falconer, Fractal Geometry, Wiley, Chichester (1990).zbMATHGoogle Scholar
  21. 21.
    J. E. Hutchinson, “Fractals and self-similarity,” Indiana Univ. Math. J., 30, 713–747 (1981).MathSciNetzbMATHGoogle Scholar
  22. 22.
    H. Triebel, Fractals and Spectra Related to Fourier Analysis and Functional Spaces, Birkhäuser, Basel (1997).zbMATHGoogle Scholar
  23. 23.
    A. F. Turbin and N. V. Pratsevityi, Fractal Sets, Functions, and Distributions [in Russian], Naukova Dumka, Kiev (1992).Google Scholar
  24. 24.
    M. V. Prats’ovytyi, Fractal Approach to the Investigation of Singular Distributions [in Ukrainian], National Pedagog. Univ., Kyiv (1998).Google Scholar
  25. 25.
    V. D. Koshmanenko, “Reconstruction of the spectral type of limiting distributions in dynamical conflict systems,” Ukr. Mat. Zh., 59, No. 6, 771–784 (2007); English translation : Ukr. Math. J., 59, No. 6, 841–857 (2007).Google Scholar
  26. 26.
    T. Karataieva and V. Koshmanenko, “Origination of the singular continuous spectrum in the conflict dynamical systems,” Methods Funct. Anal. Topol., 14, No. 1, 309–319 (2009).MathSciNetzbMATHGoogle Scholar
  27. 27.
    V. D. Koshmanenko, “Quasipoint spectral measures in the theory of dynamical conflict systems,” Ukr. Mat. Zh., 63, No. 2, 187–199 (2011); English translation : Ukr. Math. J., 63, No. 2, 222–235 (2011).Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • V. D. Koshmanenko
    • 1
  • V. O. Voloshyna
    • 2
    • 3
  1. 1.Institute of MathematicsUkrainian National Academy of SciencesKyivUkraine
  2. 2.Drahomanov National Pedagogic UniversityKyivUkraine
  3. 3.T. Shevchenko Kyiv National UniversityKyivUkraine

Personalised recommendations