Ergodicity criteria for non-expanding transformations of 2-adic spheres

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February 2014, 34(2): 367-377. doi: 10.3934/dcds.2014.34.367

Ergodicity criteria for non-expanding transformations of 2-adic spheres

Vladimir Anashin ^1, , Andrei Khrennikov ^2, and Ekaterina Yurova ^2,

1.	Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, Leninskiye Gory, 1-52, Moscow, 119991, GSP-1, Russian Federation
2.	International Center for Mathematical Modeling, Linnæus University, S-35195 Växjö, Sweden, Sweden

Received October 2012 Revised May 2013 Published August 2013

Abstract
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In the paper, we obtain necessary and sufficient conditions for ergodicity (with respect to the normalized Haar measure) of discrete dynamical systems $\langle f;\mathbf S_{2^-r}(a)\rangle$ on 2-adic spheres $\mathbf S_{2^-r}(a)$ of radius $2^{-r}$, $r\ge 1$, centered at some point $a$ from the ultrametric space of 2-adic integers $\mathbb Z_2$. The map $f\colon\mathbb Z_2\to\mathbb Z_2$ is assumed to be non-expanding and measure-preserving; that is, $f$ satisfies a Lipschitz condition with a constant 1 with respect to the 2-adic metric, and $f$ preserves a natural probability measure on $\mathbb Z_2$, the Haar measure $\mu_2$ on $\mathbb Z_2$ which is normalized so that $\mu_2(\mathbb Z_2)=1$.

Keywords: p-adic analysis., 2-adic sphere, Ergodic theory, 1-Lipschitz dynamics.

Mathematics Subject Classification: Primary: 37P20; Secondary: 11S8.

Citation: Vladimir Anashin, Andrei Khrennikov, Ekaterina Yurova. Ergodicity criteria for non-expanding transformations of 2-adic spheres. Discrete and Continuous Dynamical Systems, 2014, 34 (2) : 367-377. doi: 10.3934/dcds.2014.34.367

References:

[1]	S. Albeverio, A. Khrennikov and P. E. Kloeden, Memory retrieval as a p-adic dynamical system, Biosystems, 49 (1999), 105-115. doi: 10.1016/S0303-2647(98)00035-5.
[2]	S. Al'beverio, A. Khrennikov, B. Tirotstsi and S. de Shmedt, $p$-adic dynamical systems, Theor. Math. Phys., 114 (1998), 276-287. doi: 10.1007/BF02575441.
[3]	V. Anashin and A. Khrennikov, "Applied Algebraic Dynamics," de Gruyter Expositions in Mathematics, 49, Walter de Gruyter & Co., Berlin, 2009. doi: 10.1515/9783110203011.
[4]	V. Anashin, Uniformly distributed sequences of p-adic integers, Math. Notes, 55 (1994), 109-133. doi: 10.1007/BF02113290.
[5]	V. S. Anashin, Uniformly distributed sequences in computer algebra or how to construct program generators of random numbers, J. Math. Sci., 89 (1998), 1355-1390. doi: 10.1007/BF02355442.
[6]	V. Anashin, Uniformly distributed sequences of $p$-adic integers, Discrete Math. Appl., 12 (2002), 527-590.
[7]	V. Anashin, Ergodic transformations in the space of $p$-adic integers, in "$p$-adic Mathematical Physics" (eds. A. Yu. Khrennikov, Zoran Rakić and I. V. Volovich), AIP Conf. Proc., 826, American Institute of Physics, Melville, New York, (2006), 3-24. doi: 10.1063/1.2193107.
[8]	V. Anashin, Non-Archimedean theory of T-functions, in "Boolean Functions in Cryptology and Information Security," NATO Sci. Peace Secur. Ser. D Inf. Commun. Secur., 18, IOS, Amsterdam, (2008), 33-57. doi: 10.3233/978-1-58603-878-6-33.
[9]	V. Anashin, Non-Archimedean ergodic theory and pseudorandom generators, The Computer Journal, 53 (2010), 370-392. doi: 10.1093/comjnl/bxm101.
[10]	V. Anashin, Automata finiteness criterion in terms of van der Put series of automata functions, $p$-Adic Numbers Ultrametric Analysis and Applications, 4 (2012), 151-160. doi: 10.1134/S2070046612020070.
[11]	V. S. Anashin, A. Yu. Khrennikov and E. I. 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.
[12]	V. Anashin, A. Khrennikov and E. Yurova, Using van der Put basis to determine if a 2-adic function is measure-preserving or ergodic w.r.t. Haar measure, in "Advances in Non-Archimedean Analysis," Contemporary Mathematics, 551, American Mathematical Society, Providence, RI, (2011), 33-38. doi: 10.1090/conm/551/10883.
[13]	V. Anashin, A. Khrennikov and E. Yurova, T-functions revisited: New criteria for bijectivity/transitivity, Designes, Codes and Cryptography, (2012). doi: 10.1007/s10623-012-9741-z.
[14]	D. K. Arrowsmith and F. Vivaldi, Some p-adic representations of the Smale horseshoe, Phys. Lett. A, 176 (1993), 292-294. doi: 10.1016/0375-9601(93)90920-U.
[15]	D. K. Arrowsmith and F. Vivaldi, Geometry of p-adic Siegel discs, Physica D, 71 (1994), 222-236. doi: 10.1016/0167-2789(94)90191-0.
[16]	R. Benedetto, $p$-adic dynamics and Sullivans no wandering domain theorem, Compos. Math., 122 (2000), 281-298. doi: 10.1023/A:1002067315057.
[17]	R. Benedetto, Hyperbolic maps in $p$-adic dynamics, Ergod. Theory and Dyn. Sys., 21 (2001), 1-11. doi: 10.1017/S0143385701001043.
[18]	R. Benedetto, Components and periodic points in non-Archimedean dynamics, Proc. London Math. Soc. (3), 84 (2002), 231-256. doi: 10.1112/plms/84.1.231.
[19]	R. Benedetto, Heights and preperiodic points of polynomials over function fields, Int. Math. Res. Notices, 2005, 3855-3866. doi: 10.1155/IMRN.2005.3855.
[20]	J.-L. Chabert, A.-H. Fan and Y. Fares, Minimal dynamical systems on a discrete valuation domain, Discrete and Continuous Dynamical Systems, 25 (2009), 777-795. doi: 10.3934/dcds.2009.25.777.
[21]	Z. Coelho and W. Parry, Ergodicity of p-adic multiplication and the distribution of Fibonacci numbers, in "Topology, Ergodic Theory, Real Algebraic Geometry," Amer. Math. Soc. Transl. Ser. 2, 202, American Mathematical Society, Providence, RI, (2001), 51-70.
[22]	A.-H. Fan, M.-T. Li, J.-Y. Yao and D. Zhou, $p$-adic affine dynamical systems and applications, C. R. Acad. Sci. Paris, 342 (2006), 129-134. doi: 10.1016/j.crma.2005.11.017.
[23]	A.-H. Fan, M.-T. Li, J.-Y. Yao and D. Zhou, Strict ergodicity of affine $p$-adic dynamical systems on $\mathbbZ_p$, Adv. Math., 214 (2007), 666-700. doi: 10.1016/j.aim.2007.03.003.
[24]	A.-H. Fan, L. Liao, Y.-F. Wang and D. Zhou, $p$-adic repellers in $\mathbb Q_p$ are subshifts of finite type, C. R. Math. Acad. Sci. Paris, 344 (2007), 219-224. doi: 10.1016/j.crma.2006.12.007.
[25]	C. Favre and J. Rivera-Letelier, Theorème d'equidistribution de Brolin en dynamique $p$-adique, C. R. Math. Acad. Sci. Paris, 339 (2004), 271-276. doi: 10.1016/j.crma.2004.06.023.
[26]	M. Gundlach, A. Khrennikov and K.-O. Lindahl, On ergodic behaviour of $p$-adic dynamical systems, Infinite Dimensional Analysis, Quantum Prob. and Related Top., 4 (2001), 569-577. doi: 10.1142/S0219025701000632.
[27]	M. Gundlach, A. Khrennikov and K.-O. Lindahl, Topological transitivity for $p$-adic dynamical systems, in "$p$-adic Functional Analysis" (Ioannina, 2000), Lecture Notes in Pure and Applied Mathematics, 222, Dekker, New York, (2001), 127-132.
[28]	A. Khrennikov and M. Nilsson, "$p$-adic Deterministic and Random Dynamics," Mathematics and its Applications, 574, Kluwer Academic Publishers, Dordrecht, 2004.
[29]	J. Kingsbery, A. Levin, A. Preygel and C. E. Silva, On measure-preserving $C^1$ transformations of compact-open subsets of non-Archimedean local fields, Trans. Amer. Math. Soc., 361 (2009), 61-85. doi: 10.1090/S0002-9947-08-04686-2.
[30]	J. Kingsbery, A. Levin, A. Preygel and C. E. Silva, Dynamics of the $p$-adic shift and applications, Discrete and Continuoius Dynamical Systems, 30 (2011), 209-218. doi: 10.3934/dcds.2011.30.209.
[31]	M. V. Larin, Transitive polynomial transformations of residue rings, Discrete Math. Appl., 12 (2002), 127-140.
[32]	D.-D. Lin, T. Shi and Z.-F. Yang, Ergodic theory over $\mathbb F_2[[X]]$, Finite Fields Appl., 18 (2012), 473-491. doi: 10.1016/j.ffa.2011.11.001.
[33]	K.-O. Lindahl, On Siegel's linearization theorem for fields of prime characteristic, Nonlinearity, 17 (2004), 745-763. doi: 10.1088/0951-7715/17/3/001.
[34]	K. Mahler, "$p$-adic Numbers and their Functions," Second edition, Cambridge Tracts in Mathematics, 76, Cambridge Univ. Press, Cambridge-New York, 1981.
[35]	M. van der Put, Algèbres de fonctions continues $p$-adiques. II, (French) Nederl. Akad. Wetensch. Proc. Ser. A 71=Indag. Math., 30 (1968), 412-420
[36]	J. Rivera-Letelier, "Dynamique des Fonctions Rationelles sur des Corps Locaux," Ph.D thesis, Orsay, 2000.
[37]	J. Rivera-Letelier, Dynamique des fonctions rationelles sur des corps locaux, Asterisque, 287 (2003), 147-230.
[38]	J. Rivera-Letelier, Espace hyperbolique $p$-adique et dynamique des fonctions rationelles, Compos. Math., 138 (2003), 199-231. doi: 10.1023/A:1026136530383.
[39]	S. De Smedt and A. Khrennikov, A $p$-adic behaviour of dynamical systems, Rev. Mat. Complut., 12 (1999), 301-323.
[40]	W. H. Schikhof, "Ultrametric Calculus. An Introduction to $p$-adic Analysis," Cambridge Studies in Advanced Mathematics, 4, Cambridge University Press, Cambridge, 1984.
[41]	J. H. Silverman, "The Arithmetic of Dynamical Systems," Graduate Texts in Mathematics, 241, Springer, New York, 2007. doi: 10.1007/978-0-387-69904-2.
[42]	F. Vivaldi, The arithmetic of discretized rotations, in "$p$-adic Mathematical Physics" (eds. A. Yu. Khrennikov, Zoran Rakić and I. V. Volovich), AIP Conference Proceedings, 826, American Institute of Physics, Melville, New York, (2006), 162-173. doi: 10.1063/1.2193120.
[43]	F. Vivaldi and I. Vladimirov, Pseudo-randomness of round-off errors in discretized linear maps on the plane, Int. J. of Bifurcations and Chaos Appl. Sci. Engrg., 13 (2003), 3373-3393. doi: 10.1142/S0218127403008557.
[44]	F. Vivaldi, Algebraic and arithmetic dynamics bibliographical database. Available from: http://www.maths.qmul.ac.uk/~fv/database/algdyn.pdf.
[45]	E. I. Yurova, Van der Put basis and $p$-adic dynamics, $p$-Adic Numbers, Ultrametric Analysis and Applications, 2 (2010), 175-178. doi: 10.1134/S207004661002007X.

show all references

References:

[1]	S. Albeverio, A. Khrennikov and P. E. Kloeden, Memory retrieval as a p-adic dynamical system, Biosystems, 49 (1999), 105-115. doi: 10.1016/S0303-2647(98)00035-5.
[2]	S. Al'beverio, A. Khrennikov, B. Tirotstsi and S. de Shmedt, $p$-adic dynamical systems, Theor. Math. Phys., 114 (1998), 276-287. doi: 10.1007/BF02575441.
[3]	V. Anashin and A. Khrennikov, "Applied Algebraic Dynamics," de Gruyter Expositions in Mathematics, 49, Walter de Gruyter & Co., Berlin, 2009. doi: 10.1515/9783110203011.
[4]	V. Anashin, Uniformly distributed sequences of p-adic integers, Math. Notes, 55 (1994), 109-133. doi: 10.1007/BF02113290.
[5]	V. S. Anashin, Uniformly distributed sequences in computer algebra or how to construct program generators of random numbers, J. Math. Sci., 89 (1998), 1355-1390. doi: 10.1007/BF02355442.
[6]	V. Anashin, Uniformly distributed sequences of $p$-adic integers, Discrete Math. Appl., 12 (2002), 527-590.
[7]	V. Anashin, Ergodic transformations in the space of $p$-adic integers, in "$p$-adic Mathematical Physics" (eds. A. Yu. Khrennikov, Zoran Rakić and I. V. Volovich), AIP Conf. Proc., 826, American Institute of Physics, Melville, New York, (2006), 3-24. doi: 10.1063/1.2193107.
[8]	V. Anashin, Non-Archimedean theory of T-functions, in "Boolean Functions in Cryptology and Information Security," NATO Sci. Peace Secur. Ser. D Inf. Commun. Secur., 18, IOS, Amsterdam, (2008), 33-57. doi: 10.3233/978-1-58603-878-6-33.
[9]	V. Anashin, Non-Archimedean ergodic theory and pseudorandom generators, The Computer Journal, 53 (2010), 370-392. doi: 10.1093/comjnl/bxm101.
[10]	V. Anashin, Automata finiteness criterion in terms of van der Put series of automata functions, $p$-Adic Numbers Ultrametric Analysis and Applications, 4 (2012), 151-160. doi: 10.1134/S2070046612020070.
[11]	V. S. Anashin, A. Yu. Khrennikov and E. I. 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.
[12]	V. Anashin, A. Khrennikov and E. Yurova, Using van der Put basis to determine if a 2-adic function is measure-preserving or ergodic w.r.t. Haar measure, in "Advances in Non-Archimedean Analysis," Contemporary Mathematics, 551, American Mathematical Society, Providence, RI, (2011), 33-38. doi: 10.1090/conm/551/10883.
[13]	V. Anashin, A. Khrennikov and E. Yurova, T-functions revisited: New criteria for bijectivity/transitivity, Designes, Codes and Cryptography, (2012). doi: 10.1007/s10623-012-9741-z.
[14]	D. K. Arrowsmith and F. Vivaldi, Some p-adic representations of the Smale horseshoe, Phys. Lett. A, 176 (1993), 292-294. doi: 10.1016/0375-9601(93)90920-U.
[15]	D. K. Arrowsmith and F. Vivaldi, Geometry of p-adic Siegel discs, Physica D, 71 (1994), 222-236. doi: 10.1016/0167-2789(94)90191-0.
[16]	R. Benedetto, $p$-adic dynamics and Sullivans no wandering domain theorem, Compos. Math., 122 (2000), 281-298. doi: 10.1023/A:1002067315057.
[17]	R. Benedetto, Hyperbolic maps in $p$-adic dynamics, Ergod. Theory and Dyn. Sys., 21 (2001), 1-11. doi: 10.1017/S0143385701001043.
[18]	R. Benedetto, Components and periodic points in non-Archimedean dynamics, Proc. London Math. Soc. (3), 84 (2002), 231-256. doi: 10.1112/plms/84.1.231.
[19]	R. Benedetto, Heights and preperiodic points of polynomials over function fields, Int. Math. Res. Notices, 2005, 3855-3866. doi: 10.1155/IMRN.2005.3855.
[20]	J.-L. Chabert, A.-H. Fan and Y. Fares, Minimal dynamical systems on a discrete valuation domain, Discrete and Continuous Dynamical Systems, 25 (2009), 777-795. doi: 10.3934/dcds.2009.25.777.
[21]	Z. Coelho and W. Parry, Ergodicity of p-adic multiplication and the distribution of Fibonacci numbers, in "Topology, Ergodic Theory, Real Algebraic Geometry," Amer. Math. Soc. Transl. Ser. 2, 202, American Mathematical Society, Providence, RI, (2001), 51-70.
[22]	A.-H. Fan, M.-T. Li, J.-Y. Yao and D. Zhou, $p$-adic affine dynamical systems and applications, C. R. Acad. Sci. Paris, 342 (2006), 129-134. doi: 10.1016/j.crma.2005.11.017.
[23]	A.-H. Fan, M.-T. Li, J.-Y. Yao and D. Zhou, Strict ergodicity of affine $p$-adic dynamical systems on $\mathbbZ_p$, Adv. Math., 214 (2007), 666-700. doi: 10.1016/j.aim.2007.03.003.
[24]	A.-H. Fan, L. Liao, Y.-F. Wang and D. Zhou, $p$-adic repellers in $\mathbb Q_p$ are subshifts of finite type, C. R. Math. Acad. Sci. Paris, 344 (2007), 219-224. doi: 10.1016/j.crma.2006.12.007.
[25]	C. Favre and J. Rivera-Letelier, Theorème d'equidistribution de Brolin en dynamique $p$-adique, C. R. Math. Acad. Sci. Paris, 339 (2004), 271-276. doi: 10.1016/j.crma.2004.06.023.
[26]	M. Gundlach, A. Khrennikov and K.-O. Lindahl, On ergodic behaviour of $p$-adic dynamical systems, Infinite Dimensional Analysis, Quantum Prob. and Related Top., 4 (2001), 569-577. doi: 10.1142/S0219025701000632.
[27]	M. Gundlach, A. Khrennikov and K.-O. Lindahl, Topological transitivity for $p$-adic dynamical systems, in "$p$-adic Functional Analysis" (Ioannina, 2000), Lecture Notes in Pure and Applied Mathematics, 222, Dekker, New York, (2001), 127-132.
[28]	A. Khrennikov and M. Nilsson, "$p$-adic Deterministic and Random Dynamics," Mathematics and its Applications, 574, Kluwer Academic Publishers, Dordrecht, 2004.
[29]	J. Kingsbery, A. Levin, A. Preygel and C. E. Silva, On measure-preserving $C^1$ transformations of compact-open subsets of non-Archimedean local fields, Trans. Amer. Math. Soc., 361 (2009), 61-85. doi: 10.1090/S0002-9947-08-04686-2.
[30]	J. Kingsbery, A. Levin, A. Preygel and C. E. Silva, Dynamics of the $p$-adic shift and applications, Discrete and Continuoius Dynamical Systems, 30 (2011), 209-218. doi: 10.3934/dcds.2011.30.209.
[31]	M. V. Larin, Transitive polynomial transformations of residue rings, Discrete Math. Appl., 12 (2002), 127-140.
[32]	D.-D. Lin, T. Shi and Z.-F. Yang, Ergodic theory over $\mathbb F_2[[X]]$, Finite Fields Appl., 18 (2012), 473-491. doi: 10.1016/j.ffa.2011.11.001.
[33]	K.-O. Lindahl, On Siegel's linearization theorem for fields of prime characteristic, Nonlinearity, 17 (2004), 745-763. doi: 10.1088/0951-7715/17/3/001.
[34]	K. Mahler, "$p$-adic Numbers and their Functions," Second edition, Cambridge Tracts in Mathematics, 76, Cambridge Univ. Press, Cambridge-New York, 1981.
[35]	M. van der Put, Algèbres de fonctions continues $p$-adiques. II, (French) Nederl. Akad. Wetensch. Proc. Ser. A 71=Indag. Math., 30 (1968), 412-420
[36]	J. Rivera-Letelier, "Dynamique des Fonctions Rationelles sur des Corps Locaux," Ph.D thesis, Orsay, 2000.
[37]	J. Rivera-Letelier, Dynamique des fonctions rationelles sur des corps locaux, Asterisque, 287 (2003), 147-230.
[38]	J. Rivera-Letelier, Espace hyperbolique $p$-adique et dynamique des fonctions rationelles, Compos. Math., 138 (2003), 199-231. doi: 10.1023/A:1026136530383.
[39]	S. De Smedt and A. Khrennikov, A $p$-adic behaviour of dynamical systems, Rev. Mat. Complut., 12 (1999), 301-323.
[40]	W. H. Schikhof, "Ultrametric Calculus. An Introduction to $p$-adic Analysis," Cambridge Studies in Advanced Mathematics, 4, Cambridge University Press, Cambridge, 1984.
[41]	J. H. Silverman, "The Arithmetic of Dynamical Systems," Graduate Texts in Mathematics, 241, Springer, New York, 2007. doi: 10.1007/978-0-387-69904-2.
[42]	F. Vivaldi, The arithmetic of discretized rotations, in "$p$-adic Mathematical Physics" (eds. A. Yu. Khrennikov, Zoran Rakić and I. V. Volovich), AIP Conference Proceedings, 826, American Institute of Physics, Melville, New York, (2006), 162-173. doi: 10.1063/1.2193120.
[43]	F. Vivaldi and I. Vladimirov, Pseudo-randomness of round-off errors in discretized linear maps on the plane, Int. J. of Bifurcations and Chaos Appl. Sci. Engrg., 13 (2003), 3373-3393. doi: 10.1142/S0218127403008557.
[44]	F. Vivaldi, Algebraic and arithmetic dynamics bibliographical database. Available from: http://www.maths.qmul.ac.uk/~fv/database/algdyn.pdf.
[45]	E. I. Yurova, Van der Put basis and $p$-adic dynamics, $p$-Adic Numbers, Ultrametric Analysis and Applications, 2 (2010), 175-178. doi: 10.1134/S207004661002007X.

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