External arguments for the chaotic bands calculation in the Mandelbrot set

Abstract

In this paper we study the chaotic bands of any Mandelbrot set hyperbolic component. We use external arguments in order to identify the hyperbolic components. If we use harmonics as a tool, we can calculate the chaotic bands. Indeed, as we clearly show here, the harmonics of the external arguments of a given hyperbolic component (gene) are the external arguments of the last appearance hyperbolic components of the chaotic bands corresponding to the gene. Likewise, we show that only if the hyperbolic component is a cardioid the infinite number of the chaotic bands of such a cardioid fills up all its chaotic region.

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 [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. Here, we will cover all types of hyperbolic components and in all the Mandelbrot set.…”