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Operating with external arguments in the Mandelbrot set antenna. (English) Zbl 1008.37028

Summary: The external argument theory of Douady and Hubbard allows us to know both the potential and the field-lines in the exterior of the Mandelbrot set. Nonetheless, there are no explicit formulae to operate with external arguments, and the external argument theory is difficult to apply. In this paper we introduce some tools in order to obtain formulae to operate with external arguments in the Mandelbrot set antenna. Thus, we introduce the harmonic tool to calculate both the external arguments of the period-doubling cascade hyperbolic components and the external arguments of the last appearance hyperbolic components. Likewise, we introduce composition rules applied to external arguments that, with the aid of the concept of heredity, allows the calculation of all the external arguments that constitutes the family tree of a given external argument.

MSC:

37F45 Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations (MSC2010)
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[1] B.B. Mandelbrot, in: R.H.G. Helleman (Ed.), Nonlinear Dynamics, Ann. NY Acad. Sci. 357 (1980) 249.
[2] Mandelbrot, B.B., Physica D, 7, 224, (1983)
[3] Douady, A.; Hubbard, J.H., CR acad. sci. Paris, 294, 123, (1982)
[4] A. Douady, J.H. Hubbard, Étude Dynamique des Polynômes Complexes, Publ. Math. d’Orsay, Part I No. 84-02, 1984; Part II No. 85-04, 1985.
[5] A. Douady, in: M. Barnsley, S.G. Demko (Eds.), Chaotic Dynamics and Fractals, Academic Press, New York, 1986, p. 155.
[6] A. Douady, in: H.-O. Peitgen, P.H. Richter, The Beauty of Fractals, Springer, Berlin, 1986, p. 161 (invited contribution).
[7] B. Branner, in: R.L. Devaney, L. Keen (Eds.), Chaos and Fractals, Proceedings of the Symposium on Applied Mathematics, AMS, Vol. 39, 1989, p. 75.
[8] H.-O. Peitgen, P.H. Richter, The Beauty of Fractals, Springer, Berlin, 1986, p. 63. · Zbl 0601.58003
[9] H.-O. Peitgen, H. Jürgens, D. Saupe, Chaos and Fractals, Springer, New York, 1992, p. 802.
[10] L. Carleson, T.W. Gamelin, Complex Dynamics, Springer, New York, 1993, p. 139. · Zbl 0782.30022
[11] W. Jung, Preprint, 2001, in preparation.
[12] W. Jung, mandel.exe, a DOS program available from http:/www.iram.rwth-aachen.de/∼jung/ or http:/www.math.sunysb.edu/.
[13] E. Lau, D. Schleicher, IMS Preprint 94-19, 1994.
[14] A. Douady, X. Buff, R.L. Devaney, P. Sentenac, in: T. Lei (Ed.), The Mandelbrot Set, Cambridge University Press, Cambridge, 2000, p. 19. · Zbl 1107.37303
[15] Lorenz, E.N., J. atmos. sci., 20, 130, (1963)
[16] Ruelle, D.; Takens, F., Commun. math. phys., 20, 167, (1971) · Zbl 0227.76084
[17] J.P. Crutchfield, K. Kaneco, in: B.-L. Hao (Ed.), Directions in Chaos, Vol. II, World Scientific, Singapore, 1987.
[18] K.R. Sreenivasan, in: S.H. Davis, J.L. Lumley (Eds.), Frontiers in Fluid Mechanics, Springer, Berlin, 1985, p. 41.
[19] Beck, C., Physica D, 125, 171, (1999)
[20] M. Misiurewicz, Z. Nitecki, Mem. Am. Math. Soc. 94 (456) (1991).
[21] Romera, M.; Pastor, G.; Montoya, F., Physica A, 232, 517, (1996)
[22] Pastor, G.; Romera, M.; Montoya, F., Chaos, solitons and fractals, 7, 565, (1996)
[23] Pastor, G.; Romera, M.; Montoya, F., Phys. rev. E, 56, 1476, (1997)
[24] Romera, M.; Pastor, G.; Álvarez, G.; Montoya, F., Phys. rev. E, 58, 7214, (1998)
[25] Pastor, G.; Romera, M.; Sanz-Martín, J.C.; Montoya, F., Physica A, 256, 369, (1998)
[26] Pastor, G.; Romera, M.; Montoya, F., Physica A, 232, 536, (1996)
[27] Pastor, G.; Romera, M.; Álvarez, G.; Montoya, F., Physica A, 292, 207, (2001)
[28] Hao, B.-L.; Zheng, W.-M., Int. J. mod. phys. B, 3, 235, (1989)
[29] W.-M. Zheng, B.-L. Hao, in: B.-L. Hao (Ed.), Experimental Study and Characterization of Chaos, World Scientific, Singapore, 1990, p. 364.
[30] J. Milnor, W. Thurston, in: J.C. Alexander (Ed.), Dynamical Systems, Springer, Berlin, 1988.
[31] Metropolis, N.; Stein, M.L.; Stein, P.R., J. comb. theory, 15, 25, (1973)
[32] R.L. Devaney (Ed.), Complex Dynamical Systems, American Mathematical Society, Providence, RI, 1994, p. 23. · Zbl 0809.00024
[33] Myrberg, P.J., Ann. acad. sci. fenn-M, 336, 3, 1, (1963)
[34] Feigenbaum, M.J., J. stat. phys., 19, 25, (1978)
[35] D. Schleicher, IMS Preprint 1997/13, 1997.
[36] J. Hubbard, D. Schleicher, in: R.L. Devaney (Ed.), Complex Dynamical Systems, American Mathematical Society, Providence, RI, 1994, p. 155.
[37] B.-L. Hao, W.-M. Zheng, Applied Symbolic Dynamics and Chaos, World Scientific, Singapore, 1998, p. 23. · Zbl 0914.58017
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