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Minimal subsystems of affine dynamics on local fields. (English) Zbl 1214.11134

Summary: We describe the dynamics of an arbitrary affine dynamical system on a local field by exhibiting all its minimal subsystems. In the special case of the field \({\mathbb{Q}_p}\) of \(p\)-adic numbers, for any non-trivial affine dynamical system, we prove that the field \({\mathbb{Q}_p}\) is decomposed into a countable number of invariant balls or spheres each of which consists of a finite number of minimal subsets. Consequently, we give a complete classification of topological conjugacy for non-trivial affine dynamics on \({\mathbb{Q}_p}\). For each given prime \(p\), there is a finite number of conjugacy classes.

MSC:

11S82 Non-Archimedean dynamical systems
37P20 Dynamical systems over non-Archimedean local ground fields
Full Text: DOI

References:

[1] Anashin V.S.: Ergodic transformations in the space of p-adic integers in p-adic mathematical physics. AIP Conference Proceedings 826, 3–24 (2006) · Zbl 1152.37301 · doi:10.1063/1.2193107
[2] Anashin V.S, Khrennikov A.: Applied algebraic dynamics, de Gruyter Expositions in Mathematics 49. Walter de Gruyter & Co., Berlin (2009)
[3] T. Budnytska, Topological conjugacy classes of affine maps, $${\(\backslash\)tt arXiv:0812.4921v1}$$ .
[4] Bryk J., Silva C.: Measurable dynamics of simple p-adic polynomials. Amer. Math. Monthly 112, 212–232 (2005) · Zbl 1089.11066 · doi:10.2307/30037439
[5] Cappell S.E., Shaneson J.L.: Linear algebra and topology. Bull. Amer. Math. Soc., New Series 1, 685–687 (1979) · Zbl 0412.57030 · doi:10.1090/S0273-0979-1979-14667-6
[6] Cappell S.E., Shaneson J.L.: Nonlinear similarity of matrices. Bull. Amer. Math. Soc., New Series 1, 899–902 (1979) · Zbl 0449.57012 · doi:10.1090/S0273-0979-1979-14688-3
[7] Cappell S.E., Shaneson J.L.: Non-linear similarity. Ann. Math. 113, 315–355 (1981) · Zbl 0477.57022 · doi:10.2307/2006986
[8] Cappell S.E., Shaneson J.L.: Nonlinear similarity and differentiability. Comm. Pure Appl. Math. 38, 697–706 (1985) · Zbl 0618.58010 · doi:10.1002/cpa.3160380603
[9] Cappell S.E., Shaneson J.L.: Non-linear similarity and linear similarity are equivariant below dimension 6. Contemp. Math. 231, 59–66 (1999) · Zbl 0935.15011 · doi:10.1090/conm/231/03352
[10] Chabert J.L., Fan A.H., Fares Y.: Minimal dynamical systems on a discrete valuation domain. Discrete Cont. Dyn. Syst. 25, 777–795 (2009) · Zbl 1185.37014 · doi:10.3934/dcds.2009.25.777
[11] Coelho Z., Parry W.: Ergodicity of p-adic multiplications and the distribution of Fibonacci numbers, Topology, Ergodic Theory, Real Algebraic Geometry. Amer. Math. Soc. Transl. Ser. 2 202, 51–70 (2001) · Zbl 0985.11008
[12] H. Diao and C. Silva, Digraph representations of rational functions over the p-adic numbers, $${\(\backslash\)tt arXiv:0909.4130v1}$$ . · Zbl 1259.37065
[13] Evrard S., Fares Y.: p-adic subsets whose factorials satisfy a generalized Legendre Formula. Bull Lond Math. Soc. 40, 37–50 (2008) · Zbl 1160.11011 · doi:10.1112/blms/bdm090
[14] A. H. Fan et al., Strict ergodicity of affine p-adic dynamical systems on $${\(\backslash\)mathbb{Z}_p}$$ . Advances in Mathematics 214 (2007), 666–700. See also A. H. Fan et al.p-adic affine dynamical systems and applications, C. R. Acad. Sci. Paris, Ser. I 342 (2006), 129–134.
[15] A. H. Fan and L. M. Liao, On minimal decomposition of p-adic polynomial dynamical systems, preprint. · Zbl 1269.37036
[16] Kuiper N.H, Robbin J.W.: Topological classification of linear endomorphisms. Invent. Math. 19, 83–106 (1973) · Zbl 0251.58008 · doi:10.1007/BF01418922
[17] Gundlach M., Khrennikov A., Lindahl K.O.: On ergodic behavior of p-adic dynamical systems. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 4, 569–577 (2001) · Zbl 1040.37005 · doi:10.1142/S0219025701000632
[18] Herman M.R., Yoccoz J.C., (1983) Generalization of some theorem of small divisors to non-Archimedean fields, Geometric Dynamics, LNM 1007, Springer-Verlag: 408–447
[19] A. Khrennikov, Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models, Kluwer, 1997. · Zbl 0920.11087
[20] Khrennikov A., Nilsson M.: On the number of cycles of p-adic dynamical systems. J. Number Theory 90, 255–264 (2001) · Zbl 0994.11039 · doi:10.1006/jnth.2001.2665
[21] A. Khrennikov and M. Nilsson, p-adic deterministic and random dynamics, Kluwer Academic Publ, 2004. · Zbl 1135.37003
[22] Kingsbery J. et al.: Measurable dynamics of maps on profinite groups. Indag. Math. 18, 561–581 (2007) · Zbl 1154.37002 · doi:10.1016/S0019-3577(07)80063-2
[23] Lubin J.: Non-Archimedean dynamical systems. Compositio Mathematica 94, 321–346 (1994) · Zbl 0843.58111
[24] J. M. Luck, P. Moussa, and M. Waldschmidt (editors), Number theory and physics, Springer Proceedings in Physics 47, Springer-Verlag, 1990. · Zbl 0702.00008
[25] Oselies R., Zieschang H.: Ergodische Eigenschaften der Automorphismen p-adischer Zahlen. Arch. Math 26, 144–153 (1975) · Zbl 0303.28019 · doi:10.1007/BF01229718
[26] Schikhof W.H., Ultrametric calculus, Cambrige University Press, 1984. · Zbl 0553.26006
[27] J. Silverman, The Arithmetic of Dynamical Systems, Springer-Verlag, 2007. · Zbl 1130.37001
[28] V. S. Vladimirov, I. V. Volovich, and E. I. Zelenov, p-adic Analysis and Mathematical Physics, Series on Soviet and East European Mathematics 1, World Scientific, (1994). · Zbl 0864.46048
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