Advertisement

On measure-preserving functions over ℤ3

  • E. Yurova
Research Articles

Abstract

This paper is devoted to (discrete) p-adic dynamical systems, an important domain of algebraic and arithmetic dynamics [31]–[41], [5]–[8]. In this note we study properties of measurepreserving dynamical systems in the case p = 3. This case differs crucially from the case p = 2. The latter was studied in the very detail in [43]. We state results on all compatible functions which preserve measure on the space of 3-adic integers, using previous work of A. Khrennikov and author of present paper, see [24]. To illustrate one of the obtained theorems we describe conditions for the 3-adic generalized polynomial to be measure-preserving on ℤ3. The generalized polynomials with integral coefficients were studied in [17, 33] and represent an important class of T-functions. In turn, it is well known that T-functions are well-used to create secure and efficient stream ciphers, pseudorandom number generators.

Key words

measure-preserving generalized polynomial van der Put basis 3-adic integers 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    S. Albeverio, A. Khrennikov and P. E. Kloeden, “Memory retrieval as a p-adic dynamical system,” BioSystems 49, 105–115 (1999).CrossRefGoogle Scholar
  2. 2.
    S. Albeverio, A. Khrennikov, B. Tirozzi and D. De Smedt, “p-Adic dynamical systems,” Theor. Math. Phys, 114, 276–287 (1998).CrossRefMATHGoogle Scholar
  3. 3.
    V. Anashin and A. Khrennikov, Applied Algebraic Dynamics, De Gruyter Expositions in Math. 49 (Walter de Gruyter, Berlin — New York, 2009).CrossRefMATHGoogle Scholar
  4. 4.
    V. S. Anashin, A. Yu. Khrennikov and E. I. Yurova, “Characterization of ergodicity of p-adic dynamical systems by using the van der Put basis,” Doklady Akad. Nauk 438(2), 151–153 (2011).MathSciNetGoogle Scholar
  5. 5.
    V. Anashin, “Uniformly distributed sequences of p-adic integers,” Math. Notes 55, 109–133 (1994).MathSciNetCrossRefGoogle Scholar
  6. 6.
    V. Anashin, “Uniformly distributed sequences of p-adic integers, II,” Discrete Math. Appl. 12, 527–590 (2002).MathSciNetMATHGoogle Scholar
  7. 7.
    V. Anashin, “Non-Archimedean analysis, T-functions and cryptography,” http://arxiv.org/abs/cs/0612038/, (2006).
  8. 8.
    V. Anashin, “Wreath products in stream cipher design,” http://arxiv.org/abs/cs/0602012/, (2006).
  9. 9.
    D. K. Arrowsmith and F. Vivaldi, “Some p-adic representations of the Smale horseshoe,” Phys. Lett. A 176, 292–294 (1993).MathSciNetCrossRefGoogle Scholar
  10. 10.
    D. K. Arrowsmith and F. Vivaldi, “Geometry of p-adic Siegel discs,” Physica D 71, 222–236 (1994).MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    R. Benedetto, “p-Adic dynamics and Sullivan’s no wandering domain theorem,” Compos. Math. 122, 281–298 (2000).MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    R. Benedetto, “Hyperbolic maps in p-adic dynamics,” Ergod. Theory Dyn. Sys. 21, 1–11 (2001).MathSciNetMATHGoogle Scholar
  13. 13.
    R. Benedetto, “Components and periodic points in non-Archimedean dynamics,” Proc. London Math. Soc. 84, 231–256 (2002).MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    R. Benedetto, “Heights and preperiodic points of polynomials over function fields,” Int. Math. Res. Notices 62, 3855–3866 (2005).MathSciNetCrossRefGoogle Scholar
  15. 15.
    J.-L. Chabert, A.-H. Fan and Y. Fares, “Minimal dynamical systems on a discrete valuation domain,” Discr. Contin. Dyn. Systems A 25, 777–795 (2009).MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Z. Coelho and W. Parry, “Ergodicity of p-adic multiplication and the distribution of Fibonacci numbers,” in Topology, Ergodic Theory, Real Algebraic Geometry 202, Amer.Math. Soc. Transl. Ser. pp. 51–70 (Amer. Math. Society, 2001).MathSciNetGoogle Scholar
  17. 17.
    Zhaopeng Dai and Zhuojun Liu, The Single Cycle T-functions, MM Research Preprints KLMM, AMSS, Academia Sinica 30, 107–121 (2011).Google Scholar
  18. 18.
    B. Dragovich, A. Khrennikov and D. Mihajlovic, “Linear fractional p-adic and adelic dynamical systems,” Rep. Math. Phys. 60(1), 55–68 (2007).MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    A.-H. Fan, M.-T. Li, J.-Y. Yao and D. Zhou, “p-Adic affine dynamical systems and applications,” C. R. Acad. Sci. Paris Ser. I 342, 129–134 (2006).MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    A.-H. Fan, M.-T. Li, J.-Y. Yao and D. Zhou, “Strict ergodicity of affine p-adic dynamical systems,” Adv. Math. 214, 666–700 (2007).MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    A.-H. Fan, L. Liao, Y. F. Wang and D. Zhou, “p-adic repellers in ℚp are subshifts of finite type,” C. R. Math. Acad. Sci. Paris 344, 219–224 (2007).MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    C. Favre and J. Rivera-Letelier, “Theor’eme equidistribution de Brolin en dynamique p-adique,” C. R. Math. Acad. Sci. Paris 339, 271–276 (2004).MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    M. Gundlach, A. Khrennikov and K.-O. Lindahl, “On ergodic behaviour of p-adic dynamical systems,” Infin. Dimens. Anal. Quantum Prob. Related Fields 4(4), 569–577 (2001).MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    A. Khrennikov and E. Yurova, “Criteria of measure-preserving for p-adic dynamical systems in terms of the van der Put basis,” submitted to J. Number Theory (January 2012).Google Scholar
  25. 25.
    A. Khrennikov and P.-A. Svensson, “Attracting points of polynomial dynamical systems in fields of p-adic numbers,” IzvestiyaMath. 71(4), 753–764 (2007).MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    A. Khrennikov, M. Nilsson and N. Mainetti, “Non-Archimedean dynamics,” in p-Adic Numbers in Number Theory, Analytic Geometry and Functional Analysis, Collection of papers in honour N. De Grande-De Kimpe and L. Van Hamme. Ed. S. Caenepeel. Bull. Belgian Math. Society 141–147 (December 2002).Google Scholar
  27. 27.
    A. Khrennikov and M. Nilsson, “Behaviour of Hensel perturbations of p-adic monomial dynamical systems,” Anal.Mathematica 29, 107–133 (2003).MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    A. Khrennikov, M. Nilsson and R. Nyqvist, “The asymptotic number of periodic points of discrete polynomial p-adic dynamical systems,” ContemporaryMath. 319, 159–166 (2003).MathSciNetGoogle Scholar
  29. 29.
    A. Yu. Khrennikov and M. Nilsson, “On the number of cycles for p-adic dynamical systems,” J. Number Theory 90, 255–264 (2001).MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    A. Khrennikov, “Small denominators in complex p-adic dynamics,” Indag. Mathem. 12(2), 177–189 (2001).MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    A. Khrennikov, Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models (Kluwer, Dordreht, 1997).CrossRefMATHGoogle Scholar
  32. 32.
    A. Khrennikov and M. Nilsson, “p-Adic deterministic and random dynamics,” (Kluwer, Dordrecht, 2004).Google Scholar
  33. 33.
    A. Klimov, Applications of T-functions in Cryptography Ph. D. thesis (Weizmann Inst. of Science, 2005).Google Scholar
  34. 34.
    K-O. Lindhal, “On Siegel’s linearization theorem for fields of prime characteristic,” Nonlinearity 17, 745–763 (2004).MathSciNetCrossRefGoogle Scholar
  35. 35.
    K. Mahler, p-Adic Numbers and Their Functions (Cambridge Univ. Press, 1981).Google Scholar
  36. 36.
    J. Rivera-Letelier, Dynamique des fonctions rationelles sur des corps locaux, Ph.D. thesis (Orsay, 2000).Google Scholar
  37. 37.
    J. Rivera-Letelier, “Dynamique des fonctions rationelles sur des corps locaux,” Asterisque 147, 147–230 (2003).MathSciNetGoogle Scholar
  38. 38.
    J. Rivera-Letelier, “Expace hyperbolique p-adique et dynamique des fonctions rationelles,” Compos. Math. 138, 199–231 (2003).MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    A. De Smedt and A. Khrennikov, “A p-adic behaviour of dynamical systems,” Rev. Mat. Comput. 12, 301–323 (1999).MATHGoogle Scholar
  40. 40.
    F. Vivaldi, “The arithmetic of discretized rotations,” in p-Adic Mathematical Physics, AIP Conference Proceedings 826, 162–173 (Melville, New York, 2006).CrossRefGoogle Scholar
  41. 41.
    F. Vivaldi and I. Vladimirov, “Pseudo-randomness of round-off errors in discretized linearmaps on the plane,” Int. J. Bifurc. Chaos 13, 3373–3393 (2003).MathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    E. I. Yurova, “Van der Put basis and p-adic dynamics,” p-Adic Numbers Ultram. Anal. Appl. 2(2), 175–178 (2010).MathSciNetCrossRefGoogle Scholar
  43. 43.
    E. Yurova, V. Anashin and A. Khrennikov, “Using van der Put basis to determine if a 2-adic function is measure-preserving or ergodic w.r.t. Haar measure,” in Contemporary Mathematics: Advances in non-Archimedean Analysis 33–38 (2011).Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2012

Authors and Affiliations

  1. 1.International Center for Mathematical Modeling in Physics, Engineering, Economics, and Cognitive ScienceSchool of Computer Science, Physics and Mathematics, Linnaeus UniversityVaxjoSweden

Personalised recommendations