Equivalence between Subshrubs and Chaotic Bands in the Mandelbrot Set

External arguments for the chaotic bands calculation in the Mandelbrot set

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 hyperboli...

Chaotic bands in the Mandelbrot set

Hyperbolic components and Misiurewicz points of the chaotic region of the Mandelbrot set are located in what we call shrubs. Each shrub has two well-differentiated parts. The first one, shrub0; corresponds to the main branch of the shrub. The second one, shrubr; corresponds to all the other branches of the shrub. In this paper, we have focused on the ordering of the hyperbolic components and Mi...

Extension of the Douady-Hubbard's Theorem on Connectedness of the Mandelbrot Set to Symmetric Polynimials

Topological structure on generalized approximation space related to n-arry relation

Classical structure of rough set theory was first formulated by Z. Pawlak in [6]. The foundation of its object classification is an equivalence binary relation and equivalence classes. The upper and lower approximation operations are two core notions in rough set theory. They can also be seenas a closure operator and an interior operator of the topology induced by an equivalence relation on a u...

Application of Gray code to the cryptanalysis of chaotic cryptosystems

Gray codes have found many applications in engineering and mathematics. In this work we explain how to apply Gray codes to the determination of the control parameter and initial point of a chaotic orbit generated by the Mandelbrot map when the kneading sequence is known. Possible applications to the cryptanalysis of a certain type of cryptosystems based on Baptista’s algorithm are also discussed.