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Criteria of ergodicity for p-adic dynamical systems in terms of coordinate functions

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  • Khrennikov, Andrei
  • Yurova, Ekaterina

Abstract

This paper is devoted to the problem of ergodicity of p-adic dynamical systems. We solved the problem of characterization of ergodicity and measure preserving for (discrete) p-adic dynamical systems for arbitrary prime p for iterations based on 1-Lipschitz functions. This problem was open since long time and only the case p=2 was investigated in details. We formulated the criteria of ergodicity and measure preserving in terms of coordinate functions corresponding to digits in the canonical expansion of p-adic numbers. (The coordinate representation can be useful, e.g., for applications to cryptography.) Moreover, by using this representation we can consider non-smooth p-adic transformations. The basic technical tools are van der Put series and usage of algebraic structure (permutations) induced by coordinate functions with partially frozen variables. We illustrate the basic theorems by presenting concrete classes of ergodic functions. As is well known, p-adic spaces have the fractal (although very special) structure. Hence, our study covers a large class of dynamical systems on fractals. Dynamical systems under investigation combine simplicity of the algebraic dynamical structure with very high complexity of behavior.

Suggested Citation

  • Khrennikov, Andrei & Yurova, Ekaterina, 2014. "Criteria of ergodicity for p-adic dynamical systems in terms of coordinate functions," Chaos, Solitons & Fractals, Elsevier, vol. 60(C), pages 11-30.
  • Handle: RePEc:eee:chsofr:v:60:y:2014:i:c:p:11-30
    DOI: 10.1016/j.chaos.2014.01.001
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    References listed on IDEAS

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    1. El Naschie, M.S., 2008. "P-Adic unification of the fundamental forces and the standard model," Chaos, Solitons & Fractals, Elsevier, vol. 38(4), pages 1011-1012.
    2. Khrennikov, A.Yu. & Kozyrev, S.V., 2007. "Genetic code on the diadic plane," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 381(C), pages 265-272.
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    Cited by:

    1. Mukhamedov, Farrukh & Khakimov, Otabek, 2016. "Phase transition and chaos: P-adic Potts model on a Cayley tree," Chaos, Solitons & Fractals, Elsevier, vol. 87(C), pages 190-196.

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