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August  2017, 37(7): 3787-3804. doi: 10.3934/dcds.2017160

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

Sangtae Jeong , and  Chunlan Li

Department of Mathematics, Inha University, Incheon, 22212, Korea

* Corresponding author

Received  August 2016 Revised  February 2017 Published  April 2017

Fund Project: 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).

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.

Keywords: p-adic integers, 1-Lipschitz, measure-preserving, ergodic, van der Put basis, Mahler basis.
Mathematics Subject Classification: 37P20 (11S80, 11S82, 37A45).
Citation: Sangtae Jeong, Chunlan Li. Measure-preservation criteria for a certain class of 1-lipschitz functions on Zp in mahler's expansion. Discrete & Continuous Dynamical Systems, 2017, 37 (7) : 3787-3804. doi: 10.3934/dcds.2017160
References:
[1]

Y. Amice, Interpolation p-adique, Bull. Soc. Math. France, 92 (1964), 117-180.   Google Scholar

[2]

V. Anashin, Uniformly distributed sequences of p-adic integers, Math. Notes, 55 (1994), 109-133.  doi: 10.1007/BF02113290.  Google Scholar

[3]

V. Anashin, Uniformly distributed sequences of p-adic integers, Discrete Math. Appl., 12 (2002), 527-590.   Google Scholar

[4]

V. Anashin, The non-Archimedean theory of discrete systems, Math. Comput. Sci., 6 (2012), 375-393.  doi: 10.1007/s11786-012-0132-7.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

V. Anashin, A. 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.  Google Scholar

[8]

V. Anashin, A. Khrennikov and E. Yurova, T-functions revisited: New criteria for bijectiv- ity/transitivity, Des. Codes Cryptogr., 71 (2014), 383-407.  doi: 10.1007/s10623-012-9741-z.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 4th edition, Clarendon, Oxford, 2008.  Google Scholar

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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.  Google Scholar

[13]

Y. Jang, S. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

D. Lin, T. 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.  Google Scholar

[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.  Google Scholar

[22] K. Mahler, p-adic Numbers and Their Functions. 2nd edition, 1981.   Google Scholar
[23]

V. V. Nekrashevich and V. I. Sushchanskii, Automata, dynamical systems, and groups, Proc. Steklov Inst. Math., 231 (2000), 128-203.   Google Scholar

[24]

V. Nobauer, Zur Theorie der Polynomtransformationen und Permutationspolynome, Math. Ann., 157 (1964), 332-342.  doi: 10.1007/BF01360874.  Google Scholar

[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.  Google Scholar

[26] W. Schikhof, Ultrametric Calculus, 2006.   Google Scholar
[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.  Google Scholar

[28]

M. van der Put, Alg‘ebres de Fonctions Continues p-adiques, Universiteit Utrecht, 1967. Google Scholar

[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.  Google Scholar

[30]

E. Yurova, On measure-preserving functions over Z3, p-Adic Numbers, Ultrametric Analysis and Applications, 4 (2012), 326-335.  doi: 10.1134/S2070046612040061.  Google Scholar

[31]

M. Zieve, Cylces of Polynomial Mappings, Ph. D thesis, University of California at Berkeley, 1996. Google Scholar

show all references

References:
[1]

Y. Amice, Interpolation p-adique, Bull. Soc. Math. France, 92 (1964), 117-180.   Google Scholar

[2]

V. Anashin, Uniformly distributed sequences of p-adic integers, Math. Notes, 55 (1994), 109-133.  doi: 10.1007/BF02113290.  Google Scholar

[3]

V. Anashin, Uniformly distributed sequences of p-adic integers, Discrete Math. Appl., 12 (2002), 527-590.   Google Scholar

[4]

V. Anashin, The non-Archimedean theory of discrete systems, Math. Comput. Sci., 6 (2012), 375-393.  doi: 10.1007/s11786-012-0132-7.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

V. Anashin, A. 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.  Google Scholar

[8]

V. Anashin, A. Khrennikov and E. Yurova, T-functions revisited: New criteria for bijectiv- ity/transitivity, Des. Codes Cryptogr., 71 (2014), 383-407.  doi: 10.1007/s10623-012-9741-z.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 4th edition, Clarendon, Oxford, 2008.  Google Scholar

[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.  Google Scholar

[13]

Y. Jang, S. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

D. Lin, T. 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.  Google Scholar

[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.  Google Scholar

[22] K. Mahler, p-adic Numbers and Their Functions. 2nd edition, 1981.   Google Scholar
[23]

V. V. Nekrashevich and V. I. Sushchanskii, Automata, dynamical systems, and groups, Proc. Steklov Inst. Math., 231 (2000), 128-203.   Google Scholar

[24]

V. Nobauer, Zur Theorie der Polynomtransformationen und Permutationspolynome, Math. Ann., 157 (1964), 332-342.  doi: 10.1007/BF01360874.  Google Scholar

[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.  Google Scholar

[26] W. Schikhof, Ultrametric Calculus, 2006.   Google Scholar
[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.  Google Scholar

[28]

M. van der Put, Alg‘ebres de Fonctions Continues p-adiques, Universiteit Utrecht, 1967. Google Scholar

[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.  Google Scholar

[30]

E. Yurova, On measure-preserving functions over Z3, p-Adic Numbers, Ultrametric Analysis and Applications, 4 (2012), 326-335.  doi: 10.1134/S2070046612040061.  Google Scholar

[31]

M. Zieve, Cylces of Polynomial Mappings, Ph. D thesis, University of California at Berkeley, 1996. Google Scholar

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