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 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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