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.
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References
S. Albeverio, A. Khrennikov and P. E. Kloeden, “Memory retrieval as a p-adic dynamical system,” BioSystems 49, 105–115 (1999).
S. Albeverio, A. Khrennikov, B. Tirozzi and D. De Smedt, “p-Adic dynamical systems,” Theor. Math. Phys, 114, 276–287 (1998).
V. Anashin and A. Khrennikov, Applied Algebraic Dynamics, De Gruyter Expositions in Math. 49 (Walter de Gruyter, Berlin — New York, 2009).
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).
V. Anashin, “Uniformly distributed sequences of p-adic integers,” Math. Notes 55, 109–133 (1994).
V. Anashin, “Uniformly distributed sequences of p-adic integers, II,” Discrete Math. Appl. 12, 527–590 (2002).
V. Anashin, “Non-Archimedean analysis, T-functions and cryptography,” http://arxiv.org/abs/cs/0612038/, (2006).
V. Anashin, “Wreath products in stream cipher design,” http://arxiv.org/abs/cs/0602012/, (2006).
D. K. Arrowsmith and F. Vivaldi, “Some p-adic representations of the Smale horseshoe,” Phys. Lett. A 176, 292–294 (1993).
D. K. Arrowsmith and F. Vivaldi, “Geometry of p-adic Siegel discs,” Physica D 71, 222–236 (1994).
R. Benedetto, “p-Adic dynamics and Sullivan’s no wandering domain theorem,” Compos. Math. 122, 281–298 (2000).
R. Benedetto, “Hyperbolic maps in p-adic dynamics,” Ergod. Theory Dyn. Sys. 21, 1–11 (2001).
R. Benedetto, “Components and periodic points in non-Archimedean dynamics,” Proc. London Math. Soc. 84, 231–256 (2002).
R. Benedetto, “Heights and preperiodic points of polynomials over function fields,” Int. Math. Res. Notices 62, 3855–3866 (2005).
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).
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).
Zhaopeng Dai and Zhuojun Liu, The Single Cycle T-functions, MM Research Preprints KLMM, AMSS, Academia Sinica 30, 107–121 (2011).
B. Dragovich, A. Khrennikov and D. Mihajlovic, “Linear fractional p-adic and adelic dynamical systems,” Rep. Math. Phys. 60(1), 55–68 (2007).
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).
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).
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).
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).
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).
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).
A. Khrennikov and P.-A. Svensson, “Attracting points of polynomial dynamical systems in fields of p-adic numbers,” IzvestiyaMath. 71(4), 753–764 (2007).
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).
A. Khrennikov and M. Nilsson, “Behaviour of Hensel perturbations of p-adic monomial dynamical systems,” Anal.Mathematica 29, 107–133 (2003).
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).
A. Yu. Khrennikov and M. Nilsson, “On the number of cycles for p-adic dynamical systems,” J. Number Theory 90, 255–264 (2001).
A. Khrennikov, “Small denominators in complex p-adic dynamics,” Indag. Mathem. 12(2), 177–189 (2001).
A. Khrennikov, Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models (Kluwer, Dordreht, 1997).
A. Khrennikov and M. Nilsson, “p-Adic deterministic and random dynamics,” (Kluwer, Dordrecht, 2004).
A. Klimov, Applications of T-functions in Cryptography Ph. D. thesis (Weizmann Inst. of Science, 2005).
K-O. Lindhal, “On Siegel’s linearization theorem for fields of prime characteristic,” Nonlinearity 17, 745–763 (2004).
K. Mahler, p-Adic Numbers and Their Functions (Cambridge Univ. Press, 1981).
J. Rivera-Letelier, Dynamique des fonctions rationelles sur des corps locaux, Ph.D. thesis (Orsay, 2000).
J. Rivera-Letelier, “Dynamique des fonctions rationelles sur des corps locaux,” Asterisque 147, 147–230 (2003).
J. Rivera-Letelier, “Expace hyperbolique p-adique et dynamique des fonctions rationelles,” Compos. Math. 138, 199–231 (2003).
A. De Smedt and A. Khrennikov, “A p-adic behaviour of dynamical systems,” Rev. Mat. Comput. 12, 301–323 (1999).
F. Vivaldi, “The arithmetic of discretized rotations,” in p-Adic Mathematical Physics, AIP Conference Proceedings 826, 162–173 (Melville, New York, 2006).
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).
E. I. Yurova, “Van der Put basis and p-adic dynamics,” p-Adic Numbers Ultram. Anal. Appl. 2(2), 175–178 (2010).
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).
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Yurova, E. On measure-preserving functions over ℤ3 . P-Adic Num Ultrametr Anal Appl 4, 326–335 (2012). https://doi.org/10.1134/S2070046612040061
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DOI: https://doi.org/10.1134/S2070046612040061