Measure-preservation criteria for a certain class of 1-lipschitz functions on Z<sub><i>p</i></sub> in mahler's expansion

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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 Z_p 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 Z_p 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 Z_p, 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 Z_p in mahler's expansion. Discrete and 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.
[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. 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.
[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.
[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. Jang, S. Jeong and C. Li, Criteria of measure-preservation for 1-Lipschitz functions on F_q [[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 F_q [[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 F_q [[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. 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.
[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. Mahler, p-adic* Numbers and Their Functions. 2^nd 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. Schikhof, Ultrametric 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, Alg‘ebres 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.

show all references

References:

[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. 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.
[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.
[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. Jang, S. Jeong and C. Li, Criteria of measure-preservation for 1-Lipschitz functions on F_q [[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 F_q [[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 F_q [[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. 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.
[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. Mahler, p-adic* Numbers and Their Functions. 2^nd 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. Schikhof, Ultrametric 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, Alg‘ebres 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.

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