Advanced Search
Article Contents
Article Contents

Measure-preservation criteria for a certain class of 1-lipschitz functions on Zp in mahler's expansion

  • * Corresponding author

    * Corresponding author 

The first author was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education(2016R1D1A1B03930281)

Abstract Full Text(HTML) Related Papers Cited by
  • In this paper, we formulate a conjecture for a measure-preservation criterion of 1-Lipschitz functions defined on the ring Zp of p-adic integers, in terms of Mahler's expansion. We then provide evidence for this conjecture in the case that p = 3, and verify that it also holds for a wider class of 1-Lipschitz functions that are everywhere differentiable on Zp, which we call $\mathcal{ B}$-functions, in the sense of Anashin.

    Mathematics Subject Classification: 37P20 (11S80, 11S82, 37A45).


    \begin{equation} \\ \end{equation}
  • 加载中
  • [1] Y. Amice, Interpolation p-adique, Bull. Soc. Math. France, 92 (1964), 117-180. 
    [2] V. Anashin, Uniformly distributed sequences of p-adic integers, Math. Notes, 55 (1994), 109-133.  doi: 10.1007/BF02113290.
    [3] V. Anashin, Uniformly distributed sequences of p-adic integers, Discrete Math. Appl., 12 (2002), 527-590. 
    [4] V. Anashin, The non-Archimedean theory of discrete systems, Math. Comput. Sci., 6 (2012), 375-393.  doi: 10.1007/s11786-012-0132-7.
    [5] V. Anashin, Quantization causes waves: Smooth finitely computable functions are affine, p-Adic Numbers, Ultrametric Analysis Appl., 7 (2015), 169-227.  doi: 10.1134/S2070046615030012.
    [6] V. Anashin and A. Khrennikov, Applied Algebraic Dynamics, De Gruyter Expositions in Mathematics, 49, Walter de Gruyter & Co., Berlin, 2009. doi: 10.1515/9783110203011.
    [7] V. AnashinA. Khrennikov and E. Yurova, Characterization of ergodicity of p-adic dynamical systems by using the van der Put basis, Doklady Mathematics, 83 (2011), 306-308.  doi: 10.1134/S1064562411030100.
    [8] V. AnashinA. Khrennikov and E. YurovaT-functions revisited: New criteria for bijectiv- ity/transitivity, Des. Codes Cryptogr., 71 (2014), 383-407.  doi: 10.1007/s10623-012-9741-z.
    [9] F. Durand and F. Paccaut, Minimal polynomial dynamics on the set of 3-adic integers, Bull. Lond. Math. Soc., 41 (2009), 302-314.  doi: 10.1112/blms/bdp003.
    [10] A. Fan and L. Liao, On minimal decomposition of p-adic polynomial dynamical systems, Adv. Math., 228 (2011), 2116-2144.  doi: 10.1016/j.aim.2011.06.032.
    [11] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 4th edition, Clarendon, Oxford, 2008.
    [12] G. Hooft, Relating the quantum mechanics of discrete systems to standard canonical quantum mechanics, Found. Phys., 44 (2014), 406-425.  doi: 10.1007/s10701-014-9788-y.
    [13] Y. JangS. Jeong and C. Li, Criteria of measure-preservation for 1-Lipschitz functions on Fq [[T]] in terms of the van der Put basis and its applications., Finite Fields and Their Applications, 37 (2016), 131-157.  doi: 10.1016/j.ffa.2015.09.007.
    [14] Y. Jang, S. Jeong and C. Li, Measure-preservation criteria of 1-Lipschitz functions on Fq [[T]] in terms of the three bases of Carlitz polynomials, digit derivatives, and digit shifts,Finite Fields Appl.(to appear). doi: 10.1016/j.ffa.2015.09.007.
    [15] S. Jeong, Characterization of ergodicity of T-adic maps on F2 [[T]] using digit derivatives basis, J. Number Theory, 133 (2013), 1846-1863.  doi: 10.1016/j.jnt.2012.11.009.
    [16] S. Jeong, Toward the ergodicity of p-adic 1-Lipschitz functions represented by the van der Put series, J. Number Theory, 133 (2013), 2874-2891.  doi: 10.1016/j.jnt.2013.02.006.
    [17] S. Jeong, Shift operators and two applications to Fq [[T]], J. Number Theory, 139 (2014), 112-137.  doi: 10.1016/j.jnt.2013.12.004.
    [18] S. Jeong, Characterization of the ergodicity of 1-Lipschitz functions on Z2 using the q-Mahler basis, J. Number Theory, 151 (2015), 116-128.  doi: 10.1016/j.jnt.2014.12.007.
    [19] A. Khrennikov and E. Yurova, Criteria of measure-preserving for p-adic dynamical systems in terms of the van der Put basis, J. Number Theory, 133 (2013), 484-491.  doi: 10.1016/j.jnt.2012.08.013.
    [20] D. LinT. Shi and Z. Yang, Ergodic theory over F2 [[T]], Finite Fields and Their Applications, 18 (2012), 473-491.  doi: 10.1016/j.ffa.2011.11.001.
    [21] K. Mahler, An interpolation series for a continuous function of a p-adic variable, J. Reine Angew. Math., 199 (1958), 23-34.  doi: 10.1515/crll.1958.199.23.
    [22] K. Mahlerp-adic Numbers and Their Functions. 2nd edition, 1981. 
    [23] V. V. Nekrashevich and V. I. Sushchanskii, Automata, dynamical systems, and groups, Proc. Steklov Inst. Math., 231 (2000), 128-203. 
    [24] V. Nobauer, Zur Theorie der Polynomtransformationen und Permutationspolynome, Math. Ann., 157 (1964), 332-342.  doi: 10.1007/BF01360874.
    [25] A. M. Robert, A Course in p-adic Analysis, Graduate Texts in Mathematics, Springer-Verlag, New York, 2000. doi: 10.1007/978-1-4757-3254-2.
    [26] W. SchikhofUltrametric Calculus, 2006. 
    [27] J. H. Silverman, The Arithmetic of Dynamical Systems, Graduate Texts in Mathematics, Springer, New York, 2007. doi: 10.1007/978-0-387-69904-2.
    [28] M. van der Put, Algebres de Fonctions Continues p-adiques, Universiteit Utrecht, 1967.
    [29] E. Yurova, Van der Put basis and p-adic dynamics, p-Adic Numbers, Ultrametric Analysis and Applications, 2 (2010), 175-178.  doi: 10.1134/S207004661002007X.
    [30] E. Yurova, On measure-preserving functions over Z3, p-Adic Numbers, Ultrametric Analysis and Applications, 4 (2012), 326-335.  doi: 10.1134/S2070046612040061.
    [31] M. Zieve, Cylces of Polynomial Mappings, Ph. D thesis, University of California at Berkeley, 1996.
  • 加载中

Article Metrics

HTML views(282) PDF downloads(65) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint