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

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

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

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 $\mu_2(\mathbb Z_2)=1$.
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:
[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]

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[3]

V. Anashin and A. Khrennikov, "Applied Algebraic Dynamics,", de Gruyter Expositions in Mathematics, 49 (2009). doi: 10.1515/9783110203011. Google Scholar

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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

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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

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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

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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