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

Vladimir Anashin; Andrei Khrennikov; Ekaterina Yurova

doi:10.3934/dcds.2014.34.367

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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 & Continuous Dynamical Systems - A, 2014, 34 (2) : 367-377. doi: 10.3934/dcds.2014.34.367

References:

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[27]	M. Gundlach, A. Khrennikov and K.-O. Lindahl, Topological transitivity for $p$-adic dynamical systems,, in, 222 (2001), 127. Google Scholar
[28]	A. Khrennikov and M. Nilsson, "$p$-adic Deterministic and Random Dynamics,", Mathematics and its Applications, 574 (2004). Google Scholar
[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. doi: 10.1090/S0002-9947-08-04686-2. Google Scholar
[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. doi: 10.3934/dcds.2011.30.209. Google Scholar
[31]	M. V. Larin, Transitive polynomial transformations of residue rings,, Discrete Math. Appl., 12 (2002), 127. Google Scholar
[32]	D.-D. Lin, T. Shi and Z.-F. Yang, Ergodic theory over $\mathbb F_2[[X]]$,, Finite Fields Appl., 18 (2012), 473. doi: 10.1016/j.ffa.2011.11.001. Google Scholar
[33]	K.-O. Lindahl, On Siegel's linearization theorem for fields of prime characteristic,, Nonlinearity, 17 (2004), 745. doi: 10.1088/0951-7715/17/3/001. Google Scholar
[34]	K. Mahler, "$p$-adic Numbers and their Functions,", Second edition, 76 (1981). Google Scholar
[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. Google Scholar
[36]	J. Rivera-Letelier, "Dynamique des Fonctions Rationelles sur des Corps Locaux,", Ph.D thesis, (2000). Google Scholar
[37]	J. Rivera-Letelier, Dynamique des fonctions rationelles sur des corps locaux,, Asterisque, 287 (2003), 147. Google Scholar
[38]	J. Rivera-Letelier, Espace hyperbolique $p$-adique et dynamique des fonctions rationelles,, Compos. Math., 138 (2003), 199. doi: 10.1023/A:1026136530383. Google Scholar
[39]	S. De Smedt and A. Khrennikov, A $p$-adic behaviour of dynamical systems,, Rev. Mat. Complut., 12 (1999), 301. Google Scholar
[40]	W. H. Schikhof, "Ultrametric Calculus. An Introduction to $p$-adic Analysis,", Cambridge Studies in Advanced Mathematics, 4 (1984). Google Scholar
[41]	J. H. Silverman, "The Arithmetic of Dynamical Systems,", Graduate Texts in Mathematics, 241 (2007). doi: 10.1007/978-0-387-69904-2. Google Scholar
[42]	F. Vivaldi, The arithmetic of discretized rotations,, in, 826 (2006), 162. doi: 10.1063/1.2193120. Google Scholar
[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. doi: 10.1142/S0218127403008557. Google Scholar
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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. doi: 10.1016/S0303-2647(98)00035-5. Google Scholar
[2]	S. Al'beverio, A. Khrennikov, B. Tirotstsi and S. de Shmedt, $p$-adic dynamical systems,, Theor. Math. Phys., 114 (1998), 276. doi: 10.1007/BF02575441. Google Scholar
[3]	V. Anashin and A. Khrennikov, "Applied Algebraic Dynamics,", de Gruyter Expositions in Mathematics, 49 (2009). doi: 10.1515/9783110203011. Google Scholar
[4]	V. Anashin, Uniformly distributed sequences of p-adic integers,, Math. Notes, 55 (1994), 109. doi: 10.1007/BF02113290. Google Scholar
[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. doi: 10.1007/BF02355442. Google Scholar
[6]	V. Anashin, Uniformly distributed sequences of $p$-adic integers,, Discrete Math. Appl., 12 (2002), 527. Google Scholar
[7]	V. Anashin, Ergodic transformations in the space of $p$-adic integers,, in, 826 (2006), 3. doi: 10.1063/1.2193107. Google Scholar
[8]	V. Anashin, Non-Archimedean theory of T-functions,, in, 18 (2008), 33. doi: 10.3233/978-1-58603-878-6-33. Google Scholar
[9]	V. Anashin, Non-Archimedean ergodic theory and pseudorandom generators,, The Computer Journal, 53 (2010), 370. doi: 10.1093/comjnl/bxm101. Google Scholar
[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. doi: 10.1134/S2070046612020070. Google Scholar
[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. doi: 10.1134/S1064562411030100. Google Scholar
[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, 551 (2011), 33. doi: 10.1090/conm/551/10883. Google Scholar
[13]	V. Anashin, A. Khrennikov and E. Yurova, T-functions revisited: New criteria for bijectivity/transitivity,, Designes, (2012). doi: 10.1007/s10623-012-9741-z. Google Scholar
[14]	D. K. Arrowsmith and F. Vivaldi, Some p-adic representations of the Smale horseshoe,, Phys. Lett. A, 176 (1993), 292. doi: 10.1016/0375-9601(93)90920-U. Google Scholar
[15]	D. K. Arrowsmith and F. Vivaldi, Geometry of p-adic Siegel discs,, Physica D, 71 (1994), 222. doi: 10.1016/0167-2789(94)90191-0. Google Scholar
[16]	R. Benedetto, $p$-adic dynamics and Sullivans no wandering domain theorem,, Compos. Math., 122 (2000), 281. doi: 10.1023/A:1002067315057. Google Scholar
[17]	R. Benedetto, Hyperbolic maps in $p$-adic dynamics,, Ergod. Theory and Dyn. Sys., 21 (2001), 1. doi: 10.1017/S0143385701001043. Google Scholar
[18]	R. Benedetto, Components and periodic points in non-Archimedean dynamics,, Proc. London Math. Soc. (3), 84 (2002), 231. doi: 10.1112/plms/84.1.231. Google Scholar
[19]	R. Benedetto, Heights and preperiodic points of polynomials over function fields,, Int. Math. Res. Notices, 2005 (): 3855. doi: 10.1155/IMRN.2005.3855. Google Scholar
[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. doi: 10.3934/dcds.2009.25.777. Google Scholar
[21]	Z. Coelho and W. Parry, Ergodicity of p-adic multiplication and the distribution of Fibonacci numbers,, in, 202 (2001), 51. Google Scholar
[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. doi: 10.1016/j.crma.2005.11.017. Google Scholar
[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. doi: 10.1016/j.aim.2007.03.003. Google Scholar
[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. doi: 10.1016/j.crma.2006.12.007. Google Scholar
[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. doi: 10.1016/j.crma.2004.06.023. Google Scholar
[26]	M. Gundlach, A. Khrennikov and K.-O. Lindahl, On ergodic behaviour of $p$-adic dynamical systems,, Infinite Dimensional Analysis, 4 (2001), 569. doi: 10.1142/S0219025701000632. Google Scholar
[27]	M. Gundlach, A. Khrennikov and K.-O. Lindahl, Topological transitivity for $p$-adic dynamical systems,, in, 222 (2001), 127. Google Scholar
[28]	A. Khrennikov and M. Nilsson, "$p$-adic Deterministic and Random Dynamics,", Mathematics and its Applications, 574 (2004). Google Scholar
[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. doi: 10.1090/S0002-9947-08-04686-2. Google Scholar
[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. doi: 10.3934/dcds.2011.30.209. Google Scholar
[31]	M. V. Larin, Transitive polynomial transformations of residue rings,, Discrete Math. Appl., 12 (2002), 127. Google Scholar
[32]	D.-D. Lin, T. Shi and Z.-F. Yang, Ergodic theory over $\mathbb F_2[[X]]$,, Finite Fields Appl., 18 (2012), 473. doi: 10.1016/j.ffa.2011.11.001. Google Scholar
[33]	K.-O. Lindahl, On Siegel's linearization theorem for fields of prime characteristic,, Nonlinearity, 17 (2004), 745. doi: 10.1088/0951-7715/17/3/001. Google Scholar
[34]	K. Mahler, "$p$-adic Numbers and their Functions,", Second edition, 76 (1981). Google Scholar
[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. Google Scholar
[36]	J. Rivera-Letelier, "Dynamique des Fonctions Rationelles sur des Corps Locaux,", Ph.D thesis, (2000). Google Scholar
[37]	J. Rivera-Letelier, Dynamique des fonctions rationelles sur des corps locaux,, Asterisque, 287 (2003), 147. Google Scholar
[38]	J. Rivera-Letelier, Espace hyperbolique $p$-adique et dynamique des fonctions rationelles,, Compos. Math., 138 (2003), 199. doi: 10.1023/A:1026136530383. Google Scholar
[39]	S. De Smedt and A. Khrennikov, A $p$-adic behaviour of dynamical systems,, Rev. Mat. Complut., 12 (1999), 301. Google Scholar
[40]	W. H. Schikhof, "Ultrametric Calculus. An Introduction to $p$-adic Analysis,", Cambridge Studies in Advanced Mathematics, 4 (1984). Google Scholar
[41]	J. H. Silverman, "The Arithmetic of Dynamical Systems,", Graduate Texts in Mathematics, 241 (2007). doi: 10.1007/978-0-387-69904-2. Google Scholar
[42]	F. Vivaldi, The arithmetic of discretized rotations,, in, 826 (2006), 162. doi: 10.1063/1.2193120. Google Scholar
[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. doi: 10.1142/S0218127403008557. Google Scholar
[44]	F. Vivaldi, Algebraic and arithmetic dynamics bibliographical database., Available from: , (). Google Scholar
[45]	E. I. Yurova, Van der Put basis and $p$-adic dynamics,, $p$-Adic Numbers, 2 (2010), 175. doi: 10.1134/S207004661002007X. Google Scholar

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