July  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

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.

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 - A, 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. 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. 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. 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. Mahler, p-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. 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, 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.

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. 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. 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. 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. Mahler, p-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. 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, 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.

[1]

Aihua Fan, Shilei Fan, Lingmin Liao, Yuefei Wang. Minimality of p-adic rational maps with good reduction. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3161-3182. doi: 10.3934/dcds.2017135

[2]

Farrukh Mukhamedov, Otabek Khakimov. Chaotic behavior of the P-adic Potts-Bethe mapping. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 231-245. doi: 10.3934/dcds.2018011

[3]

Francesco Cellarosi, Ilya Vinogradov. Ergodic properties of $k$-free integers in number fields. Journal of Modern Dynamics, 2013, 7 (3) : 461-488. doi: 10.3934/jmd.2013.7.461

[4]

Martin D. Buhmann, Slawomir Dinew. Limits of radial basis function interpolants. Communications on Pure & Applied Analysis, 2007, 6 (3) : 569-585. doi: 10.3934/cpaa.2007.6.569

[5]

Oliver Junge, Alex Schreiber. Dynamic programming using radial basis functions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4439-4453. doi: 10.3934/dcds.2015.35.4439

[6]

Sohana Jahan, Hou-Duo Qi. Regularized multidimensional scaling with radial basis functions. Journal of Industrial & Management Optimization, 2016, 12 (2) : 543-563. doi: 10.3934/jimo.2016.12.543

[7]

Shingo Takeuchi. The basis property of generalized Jacobian elliptic functions. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2675-2692. doi: 10.3934/cpaa.2014.13.2675

[8]

Zhaosheng Feng. Duffing-van der Pol-type oscillator systems. Discrete & Continuous Dynamical Systems - S, 2014, 7 (6) : 1231-1257. doi: 10.3934/dcdss.2014.7.1231

[9]

Qiaolin He, Chang Liu, Xiaoding Shi. Numerical study of phase transition in van der Waals fluid. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4519-4540. doi: 10.3934/dcdsb.2018174

[10]

James Kingsbery, Alex Levin, Anatoly Preygel, Cesar E. Silva. Dynamics of the $p$-adic shift and applications. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 209-218. doi: 10.3934/dcds.2011.30.209

[11]

Sergei A. Avdonin, Boris P. Belinskiy. On the basis properties of the functions arising in the boundary control problem of a string with a variable tension. Conference Publications, 2005, 2005 (Special) : 40-49. doi: 10.3934/proc.2005.2005.40

[12]

Peter Giesl. Construction of a global Lyapunov function using radial basis functions with a single operator. Discrete & Continuous Dynamical Systems - B, 2007, 7 (1) : 101-124. doi: 10.3934/dcdsb.2007.7.101

[13]

Najla Mohammed, Peter Giesl. Grid refinement in the construction of Lyapunov functions using radial basis functions. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2453-2476. doi: 10.3934/dcdsb.2015.20.2453

[14]

Scott W. Hansen, Rajeev Rajaram. Riesz basis property and related results for a Rao-Nakra sandwich beam. Conference Publications, 2005, 2005 (Special) : 365-375. doi: 10.3934/proc.2005.2005.365

[15]

Rolando Mosquera, Aziz Hamdouni, Abdallah El Hamidi, Cyrille Allery. POD basis interpolation via Inverse Distance Weighting on Grassmann manifolds. Discrete & Continuous Dynamical Systems - S, 2019, 12 (6) : 1743-1759. doi: 10.3934/dcdss.2019115

[16]

Oliver Jenkinson. Every ergodic measure is uniquely maximizing. Discrete & Continuous Dynamical Systems - A, 2006, 16 (2) : 383-392. doi: 10.3934/dcds.2006.16.383

[17]

Xiaoqin P. Wu, Liancheng Wang. Hopf bifurcation of a class of two coupled relaxation oscillators of the van der Pol type with delay. Discrete & Continuous Dynamical Systems - B, 2010, 13 (2) : 503-516. doi: 10.3934/dcdsb.2010.13.503

[18]

Stefan Siegmund. Normal form of Duffing-van der Pol oscillator under nonautonomous parametric perturbations. Conference Publications, 2001, 2001 (Special) : 357-361. doi: 10.3934/proc.2001.2001.357

[19]

Shu-Yi Zhang. Existence of multidimensional non-isothermal phase transitions in a steady van der Waals flow. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 2221-2239. doi: 10.3934/dcds.2013.33.2221

[20]

Zhaosheng Feng, Guangyue Gao, Jing Cui. Duffing--van der Pol--type oscillator system and its first integrals. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1377-1391. doi: 10.3934/cpaa.2011.10.1377

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (12)
  • HTML views (13)
  • Cited by (1)

Other articles
by authors

[Back to Top]