Criteria of measure-preserving for \(p\)-adic dynamical systems in terms of the van der Put basis. (English) Zbl 1278.37061
The theory of dynamical systems in the fields of \(p\)-adic numbers and their algebraic extensions is of great interest. As in the general theory of dynamical systems, problems of ergodicity and measure-preserving play fundamental roles in theory of \(p\)-adic dynamical systems. This paper is the first attempt to use the van der Put basis to examine measure-preserving discrete dynamical systems in a space of \(p\)-adic integers \(\mathbb{Z}_p\) for an arbitrary prime \(p\). The authors present a complete description of all compatible measure-preserving functions in the additive form of representation. In addition, they prove the criterion in terms of coefficients with respect to the van der Put basis determining whether a compatible function \(f: \mathbb{Z}_p \to \mathbb{Z}_p\) preserves the Haar measure.
Reviewer: Masataka Kanki (Tokyo)
MSC:
| 37P20 | Dynamical systems over non-Archimedean local ground fields |
| 39A12 | Discrete version of topics in analysis |
References:
| [1] | Albeverio, S.; Khrennikov, A.; Kloeden, P. E., Memory retrieval as a \(p\)-adic dynamical system, BioSyst., 49, 105-115 (1999) |
| [2] | Albeverio, S.; Khrennikov, A.; Tirozzi, B.; De Smedt, D., \(p\)-Adic dynamical systems, Theoret. and Math. Phys., 114, 276-287 (1998) · Zbl 0974.37018 |
| [3] | Anashin, V., Uniformly distributed sequences of \(p\)-adic integers, Math. Notes, 55, 109-133 (1994) · Zbl 0835.11031 |
| [4] | Anashin, V., Uniformly distributed sequences of \(p\)-adic integers, II, Discrete Math. Appl., 12, 6, 527-590 (2002) · Zbl 1054.11041 |
| [5] | Anashin, V., Ergodic transformations in the space of \(p\)-adic integers, \((p\)-Adic Mathematical Physics, 2-nd Intʼl Conference. \(p\)-Adic Mathematical Physics, 2-nd Intʼl Conference, Belgrade, Serbia and Montenegro, 15-21 September \(2005. p\)-Adic Mathematical Physics, 2-nd Intʼl Conference. \(p\)-Adic Mathematical Physics, 2-nd Intʼl Conference, Belgrade, Serbia and Montenegro, 15-21 September 2005, AIP Conf. Proc., vol. 826 (2006)), 3-24 · Zbl 1152.37301 |
| [6] | Anashin, V.; Khrennikov, A., Applied Algebraic Dynamics, de Gruyter Exp. Math., vol. 49 (2009), Walter de Gruyter: Walter de Gruyter Berlin-New York · Zbl 1184.37002 |
| [7] | Anashin, V. S.; Khrennikov, A. Yu.; Yurova, E. I., Characterization of ergodicity of \(p\)-adic dynamical systems by using the van der Put basis, Dokl. Akad. Nauk, 438, 2, 151-153 (2011) · Zbl 1247.37007 |
| [8] | Arrowsmith, D. K.; Vivaldi, F., Some \(p\)-adic representations of the Smale horseshoe, Phys. Lett. A, 176, 292-294 (1993) |
| [9] | Arrowsmith, D. K.; Vivaldi, F., Geometry of \(p\)-adic Siegel discs, Phys. D, 71, 222-236 (1994) · Zbl 0822.11081 |
| [10] | Benedetto, R., \(p\)-Adic dynamics and Sullivanʼs no wandering domain theorem, Compos. Math., 122, 281-298 (2000) · Zbl 0996.37055 |
| [11] | Benedetto, R., Hyperbolic maps in \(p\)-adic dynamics, Ergodic Theory Dynam. Systems, 21, 1-11 (2001) · Zbl 0972.37027 |
| [12] | Benedetto, R., Components and periodic points in non-Archimedean dynamics, Proc. Lond. Math. Soc., 84, 231-256 (2002) · Zbl 1012.37005 |
| [13] | Chabert, J.-L.; Fan, A.-H.; Fares, Y., Minimal dynamical systems on a discrete valuation domain, Discrete Contin. Dyn. Syst. Ser. A, 25, 777-795 (2009) · Zbl 1185.37014 |
| [14] | Coelho, Z.; Parry, W., Ergodicity of \(p\)-adic multiplication and the distribution of Fibonacci numbers, (Topology, Ergodic Theory, Real Algebraic Geometry. Topology, Ergodic Theory, Real Algebraic Geometry, Amer. Math. Soc. Transl. Ser., vol. 202 (2001)), 51-70 · Zbl 0985.11008 |
| [15] | De Smedt, S., Orthonormal bases for \(p\)-adic continuous and continuously differentiable functions, Ann. Math. Blaise Pascal, 2, 1, 275-282 (1995) · Zbl 0830.46070 |
| [16] | De Smedt, A.; Khrennikov, A., A \(p\)-adic behaviour of dynamical systems, Rev. Mat. Complut., 12, 301-323 (1999) · Zbl 0965.37067 |
| [17] | Fan, A.-H.; Li, M.-T.; Yao, J.-Y.; Zhou, D., \(p\)-Adic affine dynamical systems and applications, C. R. Acad. Sci. Paris Ser. I, 342, 129-134 (2006) · Zbl 1081.37003 |
| [18] | Fan, A.-H.; Li, M.-T.; Yao, J.-Y.; Zhou, D., Strict ergodicity of affine \(p\)-adic dynamical systems, Adv. Math., 214, 666-700 (2007) · Zbl 1121.37010 |
| [19] | Fan, A.-H.; Liao, L.; Wang, Y. F.; Zhou, D., \(p\)-Adic repellers in Qp are subshifts of finite type, C. R. Math. Acad. Sci. Paris, 344, 219-224 (2007) · Zbl 1108.37016 |
| [20] | Favre, C.; Rivera-Letelier, J., Théorème dʼéquidistribution de Brolin en dynamique \(p\)-adique, C. R. Math. Acad. Sci. Paris, 339, 271-276 (2004) · Zbl 1052.37039 |
| [21] | Khrennikov, A.; Nilsson, M., \(p\)-Adic Deterministic and Random Dynamics (2004), Kluwer: Kluwer Dordrecht · Zbl 1135.37003 |
| [22] | Lindhal, K.-O., On Siegelʼs linearization theorem for fields of prime characteristic, Nonlinearity, 17, 745-763 (2004) · Zbl 1046.37032 |
| [23] | Mahler, K., \(p\)-Adic Numbers and Their Functions (1981), Cambridge Univ. Press · Zbl 0444.12013 |
| [24] | M. van der Put, Algèbres de fonctions continues \(p\); M. van der Put, Algèbres de fonctions continues \(p\) · Zbl 0167.43503 |
| [25] | J. Rivera-Letelier, Dynamique des fonctions rationelles sur des corps locaux, PhD thesis, Orsay, 2000.; J. Rivera-Letelier, Dynamique des fonctions rationelles sur des corps locaux, PhD thesis, Orsay, 2000. |
| [26] | Rivera-Letelier, J., Dynamique des fonctions rationelles sur des corps locaux, Astérisque, 147, 147-230 (2003) · Zbl 1140.37336 |
| [27] | Rivera-Letelier, J., Espace hyperbolique \(p\)-adique et dynamique des fonctions rationelles, Compos. Math., 138, 199-231 (2003) · Zbl 1041.37021 |
| [28] | Schikhof, W. H., Ultrametric Calculus. An Introduction to \(p\)-Adic Analysis (1984), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0553.26006 |
| [29] | Silverman, J. H., The Arithmetic of Dynamical Systems, Grad. Texts in Math., vol. 241 (2007) · Zbl 1130.37001 |
| [30] | Vivaldi, F., Algebraic and arithmetic dynamics · Zbl 0706.58039 |
| [31] | Vivaldi, F., The arithmetic of discretized rotations, (Khrennikov, A. Y.; Rakic, Z.; Volovich, I. V., \(p\)-Adic Mathematical Physics. \(p\)-Adic Mathematical Physics, AIP Conf. Proc., vol. 826 (2006), Amer. Inst. Phys.: Amer. Inst. Phys. Melville, NY), 162-173 · Zbl 1152.37329 |
| [32] | Vivaldi, F.; Vladimirov, I., Pseudo-randomness of round-off errors in discretized linear maps on the plane, Internat. J. Bifur. Chaos, 13, 3373-3393 (2003) · Zbl 1057.37044 |
| [33] | Yurova, E. I., Van der Put basis and \(p\)-adic dynamics, \(p\)-Adic Numbers, Ultrametric Analysis, and Applications, 2, 2, 175-178 (2010) · Zbl 1250.37037 |
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