Operating with external arguments in the Mandelbrot set antenna

Abstract

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. 1

A Method to Solve the Limitations in Drawing External Rays of the Mandelbrot Set

“…We are interested in the drawing of external rays of M [9][10][11][12][13][14][15][16][17][18], and in this paper we will analyze the possibilities of the new quadruple 128 bits format of the IEEE 754-2008 Standard. Unfortunately, as we will see in Section 5, the resolution of the quadruple format is not sufficient in some of the cases, and the same occurs with a hypothetical octuple 256 bits format (not defined yet).…”

Operating with External Arguments of Douady and Hubbard

“…Other ways of calculating external arguments are the well-known tuning algorithm and the algorithm to calculate the external arguments of a notable point, both of them of Douady [11]. To finish, let us say that harmonics provide a compact tool to calculate the external arguments of the structural components [12,13] of both the Mandelbrot set main antenna [5] and the shrub 0 of chaotic bands [6].…”

“…In two previous papers [5,6], we studied how to operate with external arguments of Douady and Hubbard [7]. In [5], we only studied the external arguments of a part of the hyperbolic components, the structural components [8], and only in the Mandelbrot set antenna. In [6], we studied the external arguments of the same structural components, but this time in the chaotic bands of the Mandelbrot set.…”

Equivalence between subshrubs and chaotic bands in the Mandelbrot set

“…Likewise it shows the correspondence between each hyperbolic component of the period-doubling cascade G i (i = 0,1,2,...) and the chaotic band B i (i = 0,1,2,...). Calculations, that in [15] were done for symbolic sequences, in [10] have been done for external arguments. As can be seen in Figure 2.1, the Myrberg-Feigenbaum point can be considered as a "diabolic mirror" that makes to correspond the main cardioid G 0 , which is the last hyperbolic component on the right and has period 1, with B 0 , which is the last chaotic band on the left and has period 1; the first disc of the period doubling cascade G 1 , which is the penultimate hyperbolic component on the right and has period 2, with B 1 , which is the penultimate chaotic band on the left and has period 2; and so on.…”