Skip to main content
SpringerLink
Log in

Recurrence equations over trees in a non-Archimedean context

  • Research Articles
  • Published:
P-Adic Numbers, Ultrametric Analysis, and Applications Aims and scope Submit manuscript

Abstract

In the present paper we study recurrence equations over k-ary trees. Namely, each equation is assigned to a vertex of the tree, and they are generated by contractive functions defined on an arbitrary non-Archimedean algebra. The main result of this paper states that the given equations have at most one solution. Moreover, we also provide the existence of unique solution of the equations. We should stress that the non-Archimedeanity of the algebra is essentially used, therefore, the methods applied in the present paper are not valid in the Archimedean setting.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. S. Albeverio, A. Yu. Khrennikov and V. M. Shelkovich, Theory of p-Adic Distributions. Linear and Nonlinear Models (Cambridge Univ. Press, Cambridge, 2010).

    Book  MATH  Google Scholar 

  2. V. Anashin and A. Khrennikov, Applied Algebraic Dynamics (Walter de Gruyter, Berlin, New York, 2009).

    Book  MATH  Google Scholar 

  3. R. Benedetto, “Hyperbolic maps in p-adic dynamics,” Ergod. Th. Dynam. Sys. 21, 1–11 (2001).

    Article  MATH  MathSciNet  Google Scholar 

  4. B. Dragovich, A. Yu. Khrennikov, S. V. Kozyrev and I. V. Volovich, “On p-adic mathematical physics,” p-Adic Numbers Ultr. Anal. Appl. 1(1), 1–17 (2009).

    Article  MATH  MathSciNet  Google Scholar 

  5. A. Escassut, Ultrametric Banach Algebras (World Sci., Singapore, 2003).

    Book  MATH  Google Scholar 

  6. A. H. Fan, S. L Fan, L. M. Liao and Y. F. Wang, “On minimal decomposition of p-adic homographic dynamical systems,” Adv. Math. 257, 92–135 (2014).

    Article  MATH  MathSciNet  Google Scholar 

  7. O. N. Khakimov, “On p-adic Gibbs measures for Ising model with four competing interactions,” p-Adic Numbers Ultr. Anal. Appl. 5, 194–203 (2013).

    Article  MATH  MathSciNet  Google Scholar 

  8. O. N. Khakimov, “On a Generalized p-Adic Gibbs Measure for Ising Model on Trees,” p-Adic Numbers Ultr. Anal. Appl. 6, 207–218 (2014).

    Article  MathSciNet  Google Scholar 

  9. M. Khamraev and F. M. Mukhamedov, “On p-adic λ-model on the Cayley tree,” J. Math. Phys. 45, 4025–4034 (2004).

    Article  MATH  MathSciNet  Google Scholar 

  10. M. Khamraev and F. M. Mukhamedov, “On a class of rational p-adic dynamical systems,” J. Math. Anal. Appl. 315, 76–89 (2006).

    Article  MATH  MathSciNet  Google Scholar 

  11. A. Yu. Khrennikov, Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models (Kluwer Acad. Publ., Dordrecht, 1997).

    MATH  Google Scholar 

  12. A. Yu. Khrennikov, F. Mukhamedov and J. F. F. Mendes, “On p-adic Gibbs measures of countable state Potts model on the Cayley tree,” Nonlinearity 20, 2923–2937 (2007).

    Article  MATH  MathSciNet  Google Scholar 

  13. A. Yu. Khrennikov and F. Mukhamedov, “On uniqueness of Gibbs measure for p-adic countable state Potts model on the Cayley tree,” Nonlin. Anal.: Theor. Meth. Appl. 71, 5327–5331 (2009).

    Article  MATH  MathSciNet  Google Scholar 

  14. A. Yu. Khrennikov and M. Nilsson, p-Adic Deterministic and Random Dynamical Systems (Kluwer, Dordreht, 2004).

    Book  Google Scholar 

  15. N. Koblitz, p-Adic Numbers, p-Adic Analysis and Zeta-Function (Berlin, Springer, 1977).

    Book  Google Scholar 

  16. E. Marinary and G. Parisi, “On the p-adic five point function,” Phys. Lett. B 203, 52–56 (1988).

    Article  MathSciNet  Google Scholar 

  17. F. M. Mukhamedov, “On the existence of generalized Gibbs measures for the one-dimensional p-adic countable state Potts model,” Proc. Steklov Inst.Math. 265, 165–176 (2009).

    Article  MATH  MathSciNet  Google Scholar 

  18. F. Mukhamedov, “On a recursive equation over p-adic field,” Appl.Math. Lett. 20, 88–92 (2007).

    Article  MATH  MathSciNet  Google Scholar 

  19. F. Mukhamedov, “A dynamical system approach to phase transitions p-adic Potts model on the Cayley tree of order two,” Rep. Math. Phys. 70, 385–406 (2012).

    Article  MATH  MathSciNet  Google Scholar 

  20. F. Mukhamedov, “On dynamical systems and phase transitions for q + 1-state p-adic Potts model on the Cayley tree,” Math. Phys.Anal. Geom. 16, 49–87 (2013).

    Article  MATH  MathSciNet  Google Scholar 

  21. F. Mukhamedov, “On strong phase transition for one dimensional countable state p-adic Potts model,” J. Stat. Mech. P01007 (2014).

    Google Scholar 

  22. F. Mukhamedov and H. Akin, “On p-adic Potts model on the Cayley tree of order three,” Theor. Math. Phys. 176, 1267–1279 (2013).

    Article  MATH  MathSciNet  Google Scholar 

  23. F. M. Mukhamedov and U. A. Rozikov, “On Gibbs measures of p-adic Potts model on the Cayley tree,” Indag. Math. N.S. 15, 85–100 (2004).

    Article  MATH  MathSciNet  Google Scholar 

  24. F. M. Mukhamedov and U. A. Rozikov, “On inhomogeneous p-adic Potts model on a Cayley tree,” Infin. Dimens. Anal. Quantum Probab. Relat. Top. 8, 277–290 (2005).

    Article  MATH  MathSciNet  Google Scholar 

  25. C. Perez-Garcia and W. H. Schikhof, Locally Convex Spaces over Non-Archimedean Valued Fields (Cambridge Univ. Press, 2010).

    Book  MATH  Google Scholar 

  26. J. Rivera-Letelier, “Dynamics of rational functions over local fields,” Astérisque 287, 147–230 (2003).

    MathSciNet  Google Scholar 

  27. A. van Rooij, Non-Archimedean Functional Analysis (Marcel Dekker, New York, 1978).

    MATH  Google Scholar 

  28. U. A. Rozikov, GibbsMeasures on Cayley Trees (World Sci., Singapore, 2013).

    Book  Google Scholar 

  29. U. Rozikov and I. A. Sattorov, “On a nonlinear p-adic dynamical system,” p-Adic Numbers Ultr. Anal. Appl. 6, 54–65 (2014).

    Article  Google Scholar 

  30. J. H. Silverman, The Arithmetic of Dynamical Systems (Springer-Verlag, New York, 2007).

    Book  MATH  Google Scholar 

  31. V. S. Vladimirov, I. V. Volovich and E. I. Zelenov, p-Adic Analysis andMathematical Physics (World Sci., Singapore, 1994).

    Book  Google Scholar 

  32. I. V. Volovich, “Number theory as the ultimate physical theory,” p-Adic Numbers Ultr. Anal. Appl. 2, 77–87 (2010); Preprint CERN TH.4781/87 (1987).

    Article  MATH  MathSciNet  Google Scholar 

  33. I. V. Volovich, “p-Adic string,” Class. Quant. Grav. 4, L83–L87 (1987).

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computational & Theoretical Sciences Faculty of Science, International Islamic University Malaysia, P.O. Box, 141, 25710, Kuantan, Pahang, Malaysia

    F. Mukhamedov

Authors
  1. F. Mukhamedov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to F. Mukhamedov.

Additional information

The text was submitted by the author in English.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mukhamedov, F. Recurrence equations over trees in a non-Archimedean context. P-Adic Num Ultrametr Anal Appl 6, 310–317 (2014). https://doi.org/10.1134/S2070046614040062

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S2070046614040062

Key words

Search

Navigation