p-Adic Dynamical Systems of the Function ax/x2 + a

  • U. A. RozikovEmail author
  • I. A. Sattarov
  • S. Yam
Research Articles


We show that any (1, 2)-rational function with a unique fixed point is topologically conjugate to a (2, 2)-rational function or to the function f(x) = ax/x2 + a. The case (2, 2) was studied in our previous paper, here we study the dynamical systems generated by the function f on the set of complex p-adic field ℂp. We show that the unique fixed point is indifferent and therefore the convergence of the trajectories is not the typical case for the dynamical systems. We construct the corresponding Siegel disk of these dynamical systems. We determine a sufficiently small set containing the set of limit points. It is given all possible invariant spheres.We show that the p-adic dynamical system reduced on each invariant sphere is not ergodic with respect to Haar measure on the set of p-adic numbers p.Moreover some periodic orbits of the system are investigated.

Key words

rational dynamical systems fixed point invariant set Siegel disk complex p-adic field ergodic 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    S. Albeverio, U. A. Rozikov and I. A. Sattarov, “p-Adic (2, 1)-rational dynamical systems,” J. Math. Anal. Appl. 398 (2), 553–566 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    V. Anashin and A. Khrennikov, Applied Algebraic Dynamics, de Gruyter Expositions in Math. 49 (Walter de Gruyter, Berlin-New York, 2009).CrossRefGoogle Scholar
  3. 3.
    N. Koblitz, p-Adic Numbers, p-Adic Analysis and Zeta-Function (Springer, Berlin, 1977).CrossRefzbMATHGoogle Scholar
  4. 4.
    H.-O. Peitgen, H. Jungers and D. Saupe, Chaos Fractals (Springer, Heidelberg-New York, 1992).CrossRefzbMATHGoogle Scholar
  5. 5.
    A. C. M. van Rooij, Non-Archimedean Functional Analysis, Monographs and Textbooks in Pure and AppliedMath. 51 (Marcel Dekker, Inc., New York, 1978).Google Scholar
  6. 6.
    U. A. Rozikov and I. A. Sattarov, “On a non-linear p-adic dynamical system,” p-Adic Numbers Ultrametric Anal. Appl. 6 (1), 53–64 (2014).CrossRefzbMATHGoogle Scholar
  7. 7.
    U. A. Rozikov and I. A. Sattarov, “p-Adic dynamical systems of (2, 2)-rational functions with unique fixed point,” Chaos Solit. Fract. 105, 260–270 (2017).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    I. A. Sattarov, “p-Adic (3, 2)-rational dynamical systems,” p-Adic Numbers Ultrametric Anal. Appl. 7 (1), 39–55 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    W. H. Schikhof, Ultrametric Calculus: An introduction to p-adic analysis, Cambridge Studies in Advanced Math. 4 (Cambridge Univ. Press, Cambridge, 2006).Google Scholar
  10. 10.
    V. S. Vladimirov, I. V. Volovich and E. I. Zelenov, p-Adic Analysis and Mathematical Physics (World Scientific, River Edge, N. Y., 1994).CrossRefzbMATHGoogle Scholar
  11. 11.
    P. Walters, An Introduction to Ergodic Theory (Springer, Berlin-Heidelberg-New York, 1982).CrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Institute of Mathematics81, Mirzo Ulug’bek str.TashkentUzbekistan
  2. 2.California State University, Monterey BaySeasideUSA

Personalised recommendations