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On the chaotic behavior of a generalized logistic \(p\)-adic dynamical system. (English) Zbl 1128.37031

Iteration of a generalized logistic map \(x\mapsto (ax)^2(x+1)\) on the \(p\)-adic numbers is studied from a dynamical point of view. The basins of attraction and Siegel disks are described, and the geometry of the Siegel disks is related to the structure of the orbits. The arguments used exploit the ultrametric inequality and Hensel’s lemma.

MSC:

37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
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[1] Anashin, V., Ergodic transformations in the space of \(p\)-adic integers, \((p\)-Adic Mathematical Physics. \(p\)-Adic Mathematical Physics, AIP Conf. Proc., vol. 826 (2006), Melville: Melville New York), 3-24 · Zbl 1152.37301
[2] Arrowsmith, D. K.; Vivaldi, F., Geometry of \(p\)-adic Siegel discs, Phys. D, 71, 222-236 (1994) · Zbl 0822.11081
[3] Albeverio, S.; Khrennikov, A.; Kloeden, P. E., Memory retrieval as a \(p\)-adic dynamical system, BioSys., 49, 105-115 (1999)
[4] Albeverio, S.; Kloeden, P. E.; Khrennikov, A., Human memory as a \(p\)-adic dynamical system, Theoret. and Math. Phys., 117, 1414-1422 (1998) · Zbl 0948.37060
[5] Albeverio, S.; Khrennikov, A.; Tirozzi, B.; De Smedt, S., \(p\)-Adic dynamical systems, Theoret. and Math. Phys., 114, 276-287 (1998) · Zbl 0974.37018
[6] Aref’eva, I. Ya.; Dragovich, B.; Frampton, P. H.; Volovich, I. V., Wave function of the universe and \(p\)-adic gravity, Internat. J. Modern Phys. A, 6, 4341-4358 (1991) · Zbl 0733.53039
[7] Aulbach, B.; Kininger, B., An elementary proof for hyperbolicity and chaos of the logistic maps, J. Difference Equ. Appl., 10, 1243-1250 (2004) · Zbl 1062.37024
[8] Avetisov, V. A.; Bikulov, A. H.; Kozyrev, S. V.; Osipov, V. A., \(p\)-Adic models of ultrametric diffusion constrained by hierarchical energy landscapes, J. Phys. A, 35, 177-189 (2002) · Zbl 1038.82077
[9] Batra, A.; Morton, P., Algebraic dynamics of polynomial maps on the algebraic closure of a finite field II, Rocky Mountain J. Math., 24, 905-932 (1994) · Zbl 0816.11064
[10] Benedetto, R., Hyperbolic maps in \(p\)-adic dynamics, Ergodic Theory Dynam. Systems, 21, 1-11 (2001) · Zbl 0972.37027
[11] Benedetto, R., \(p\)-Adic dynamics and Sullivan’s no wandering domains theorem, Compos. Math., 122, 281-298 (2000) · Zbl 0996.37055
[12] Benedetto, R., Non-Archimedean holomorphic maps and the Ahlfors islands theorem, Amer. J. Math., 125, 581-622 (2003) · Zbl 1041.30021
[13] Bézivin, J.-P., Fractions rationelles hyperboliques \(p\)-adiques, Acta Arith., 112, 151-175 (2004) · Zbl 1065.11095
[14] Call, G.; Silverman, J., Canonical height on varieties with morphisms, Compos. Math., 89, 163-205 (1993) · Zbl 0826.14015
[15] Devaney, R. L., An Introduction to Chaotic Dynamical Systems (1989), Addison-Wesley: Addison-Wesley New York · Zbl 0695.58002
[16] Dremov, V.; Shabat, G.; Vymova, P., On the chaotic properties of quadratic maps over non-Archimedean fields, \((p\)-Adic Mathematical Physics. \(p\)-Adic Mathematical Physics, AIP Conf. Proc., vol. 826 (2006), Melville: Melville New York), 43-54 · Zbl 1152.37328
[17] Dubischer, D.; Gundlach, V. M.; Khrennikov, A.; Steinkamp, O., Attractors of random dynamical system over \(p\)-adic numbers and a model of ‘noisy’ cognitive process, Phys. D, 130, 1-12 (1999) · Zbl 1066.91599
[18] Gundlach, V. 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
[19] Freund, P. G.O.; Witten, E., Adelic string amplitudes, Phys. Lett. B, 199, 191-194 (1987)
[20] Herman, M.; Yoccoz, J.-C., Generalizations of some theorems of small divisors to non-Archimedean fields, (Geometric Dynamics. Geometric Dynamics, Rio de Janeiro, 1981. Geometric Dynamics. Geometric Dynamics, Rio de Janeiro, 1981, Lecture Notes in Math., vol. 1007 (1983), Springer: Springer Berlin), 408-447
[21] Hsia, L., Closure of periodic points over a non-archimedean field, J. London Math. Soc., 62, 685-700 (2000) · Zbl 1022.11060
[22] Jaganathan, R.; Sinha, S., A \(q\)-deformed nonlinear map, Phys. Lett. A, 338, 277-287 (2005) · Zbl 1136.82304
[23] Khamraev, M.; Mukhamedov, F. M., On a class of rational \(p\)-adic dynamical systems, J. Math. Anal. Appl., 315, 76-89 (2006) · Zbl 1085.37040
[24] Khamraev, M.; Mukhamedov, F. M.; Rozikov, U. A., On uniqueness of Gibbs measures for \(p\)-adic nonhomogeneous \(λ\)-model on the Cayley tree, Lett. Math. Phys., 70, 17-28 (2004) · Zbl 1065.46057
[25] Koblitz, N., \(p\)-Adic Numbers, \(p\)-Adic Analysis and Zeta-Function (1977), Springer: Springer Berlin · Zbl 0364.12015
[26] Khrennikov, A. Yu., \(p\)-Adic quantum mechanics with \(p\)-adic valued functions, J. Math. Phys., 32, 932-936 (1991) · Zbl 0746.46067
[27] Khrennikov, A. Yu., \(p\)-Adic Valued Distributions in Mathematical Physics (1994), Kluwer: Kluwer Dordrecht · Zbl 0833.46061
[28] Khrennikov, A. Yu., \(p\)-Adic description of chaos, (Alfinito, E.; Boti, M., Nonlinear Physics: Theory and Experiment (1996), WSP: WSP Singapore), 177-184 · Zbl 0941.82502
[29] Khrennikov, A. Yu., Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models (1997), Kluwer: Kluwer Dordrecht · Zbl 0920.11087
[30] A.Yu. Khrennikov, The description of Grain’s functioning by the \(p\)-adic dynamical system, Preprint Ruhr Univ. Bochum, SFB-237, N. 355; A.Yu. Khrennikov, The description of Grain’s functioning by the \(p\)-adic dynamical system, Preprint Ruhr Univ. Bochum, SFB-237, N. 355
[31] Khrennikov, A. Yu.; Nilsson, M., \(p\)-Adic Deterministic and Random Dynamical Systems (2004), Kluwer: Kluwer Dordrecht
[32] Lindahl, K. O., On Siegels linearization theorem for fields of prime characteristic, Nonlinearity, 17, 745-763 (2004) · Zbl 1046.37032
[33] Lubin, J., Nonarchimedean dynamical systems, Compos. Math., 94, 3, 321-346 (1994) · Zbl 0843.58111
[34] Ledrappier, F.; Pollicott, M., Distribution results for lattices in \(SL(2, Q_p)\), Bull. Braz. Math. Soc. (N.S.), 36, 2, 143-176 (2005) · Zbl 1101.37005
[35] Marinary, E.; Parisi, G., On the \(p\)-adic five point function, Phys. Lett. B, 203, 52-56 (1988)
[36] Mukhamedov, F. M., On a recursive equation over \(p\)-adic field, Appl. Math. Lett., 20, 88-92 (2007) · Zbl 1154.39001
[37] Mukhamedov, F. M.; Rozikov, U. A., On rational \(p\)-adic dynamical systems, Methods Funct. Anal. Topology, 10, 2, 21-31 (2004) · Zbl 1053.37022
[38] Mukhamedov, F. M.; Rozikov, U. A., On Gibbs measures of \(p\)-adic Potts model on the Cayley tree, Indag. Math. (N.S.), 15, 85-100 (2004) · Zbl 1161.82311
[39] Mukhamedov, F. M.; Rozikov, U. A., On inhomogeneous \(p\)-adic Potts model on a Cayley tree, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 8, 277-290 (2005) · Zbl 1096.82007
[40] Mukhamedov, F. M.; Rozikov, U. A.; Mendes, J. F.F., On phase transitions for \(p\)-adic Potts model with competing interactions on a Cayley tree, \((p\)-Adic Mathematical Physics. \(p\)-Adic Mathematical Physics, AIP Conf. Proc., vol. 826 (2006), Melville: Melville New York), 140-150 · Zbl 1152.82309
[41] Pezda, T., Polynomial cycles in certain local domains, Acta Arith., 66, 11-22 (1994) · Zbl 0803.11063
[42] Peitgen, H.-O.; Jungers, H.; Saupe, D., Chaos Fractals (1992), Springer: Springer Heidelberg
[43] Rivera-Letelier, J., Dynamics of rational functions over local fields, Astérisque, 287, 147-230 (2003) · Zbl 1140.37336
[44] Robert, A. M., A Course of \(p\)-Adic Analysis (2000), Springer: Springer New York
[45] Schikhof, W. H., Ultrametric Calculus (1984), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0553.26006
[46] Shabat, B., \(p\)-Adic entropies of logistic maps, Proc. Steklov Inst. Math., 245, 257-263 (2004) · Zbl 1098.37047
[47] Thiran, E.; Verstegen, D.; Weters, J., \(p\)-Adic dynamics, J. Stat. Phys., 54, 3/4, 893-913 (1989) · Zbl 0672.58019
[48] Vladimirov, V. S.; Volovich, I. V.; Zelenov, E. I., \(p\)-Adic Analysis and Mathematical Physics (1994), World Scientific: World Scientific Singapore · Zbl 0812.46076
[49] I.V. Volovich, Number theory as the ultimate physical theory, preprint, TH, 4781/87; I.V. Volovich, Number theory as the ultimate physical theory, preprint, TH, 4781/87 · Zbl 1258.81074
[50] Volovich, I. V., \(p\)-Adic strings, Classical Quantum Gravity, 4, L83-L87 (1987)
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