Pastor, G.; Romera, M.; Alvarez, G.; Nunez, J.; Arroyo, D.; Montoya, F. Operating with external arguments of Douady and Hubbard. (English) Zbl 1146.37032 Discrete Dyn. Nat. Soc. 2007, Article ID 45920, 17 p. (2007). Summary: The external arguments of the external rays theory of A. Douady and J. H. Hubbard [C. R. Acad. Sci., Paris, Sér. I 294, 123–125 (1982; Zbl 0483.30014)] is a valuable tool in order to analyze the Mandelbrot set, a typical case of discrete dynamical system used to study nonlinear phenomena. We suggest here a general method for the calculation of the external arguments of external rays landing at the hyperbolic components root points of the Mandelbrot set. Likewise, we present a general method for the calculation of the external arguments of external rays landing at Misiurewicz points. Cited in 3 Documents MSC: 37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets 37F45 Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations (MSC2010) 30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable Keywords:Mandelbrot set; nonlinear phenomena; external arguments of external rays; Misiurewicz points Citations:Zbl 0483.30014 PDFBibTeX XMLCite \textit{G. Pastor} et al., Discrete Dyn. Nat. Soc. 2007, Article ID 45920, 17 p. (2007; Zbl 1146.37032) Full Text: DOI EuDML References: [1] B. Branner, “The Mandelbrot set,” in Chaos and Fractals, vol. 39 of Proceedings Symposium Applied Mathematics, pp. 75-105, American Mathematical Society, Providence, RI, USA, 1989. [2] M. Misiurewicz and Z. Nitecki, “Combinatorial patterns for maps of the interval,” Memoirs of the American Mathematical Society, vol. 94, no. 456, p. 112, 1991. · Zbl 0745.58019 [3] G. Pastor, M. Romera, and F. Montoya, “On the calculation of Misiurewicz patterns in one-dimensional quadratic maps,” Physica A, vol. 232, no. 1-2, pp. 536-553, 1996. [4] M. Romera, G. Pastor, and F. Montoya, “Misiurewicz points in one-dimensional quadratic maps,” Physica A, vol. 232, no. 1-2, pp. 517-535, 1996. · Zbl 1080.37562 [5] G. Pastor, M. Romera, G. Alvarez, and F. Montoya, “Operating with external arguments in the Mandelbrot set antenna,” Physica D, vol. 171, no. 1-2, pp. 52-71, 2002. · Zbl 1008.37028 [6] G. Pastor, M. Romera, G. Alvarez, and F. Montoya, “External arguments for the chaotic bands calculation in the Mandelbrot set,” Physica A, vol. 353, pp. 145-158, 2005. [7] A. Douady and J. H. Hubbard, “Itération des polynômes quadratiques complexes,” Comptes Rendus des Séances de l’Académie des Sciences, vol. 294, no. 3, pp. 123-126, 1982. · Zbl 0483.30014 [8] G. Pastor, M. Romera, and F. Montoya, “Harmonic structure of one-dimensional quadratic maps,” Physical Review E, vol. 56, no. 2, pp. 1476-1483, 1997. [9] W. Jung, Programs for dynamical systems, http://www.iram.rwth-aachen.de/ jung/indexp.html. [10] R. L. Devaney, “The complex dynamics of quadratic polynomials,” in Complex Dynamical Systems, R. L. Devaney, Ed., vol. 49 of Proceedings Symposium Applied Mathematics, pp. 1-29, American Mathematical Society, Cincinnati, Ohio, USA, 1994. · Zbl 0820.58032 [11] A. Douady, “Algorithms for computing angles in the Mandelbrot set,” in Chaotic Dynamics and Fractals, M. Barnsley and S. G. Demko, Eds., vol. 2 of Notes and Reports in Mathematics in Science and Engineering, pp. 155-168, Academic Press, Atlanta, Ga, USA, 1986. · Zbl 0603.30030 [12] M. Romera, G. Pastor, G. Alvarez, and F. Montoya, “Shrubs in the Mandelbrot set ordering,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 13, no. 8, pp. 2279-2300, 2003. · Zbl 1057.37048 [13] G. Pastor, M. Romera, G. Alvarez, D. Arroyo, and F. Montoya, “Equivalence between subshrubs and chaotic bands in the Mandelbrot set,” Discrete Dynamics in Nature and Society, vol. 2006, Article ID 70471, 25 pages, 2006. · Zbl 1130.37377 [14] E. Lau and D. Schleicher, “Internal addresses in the Mandelbrot set and irreducibility of polynomials,” http://www.math.sunysb.edu/cgi-bin/preprint.pl?ims94-19. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.