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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 |
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.
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 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. 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. 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, 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 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. 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. 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, 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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