Discrete Dynamics in Nature and Society
Volume 2006 (2006), Issue 5, Article ID 70471, 25 pages
doi:10.1155/DDNS/2006/70471
Equivalence between subshrubs and chaotic bands in the Mandelbrot set
G. Pastor
, M. Romera
, G. Alvarez
, D. Arroyo
and F. Montoya
Departamento de Tratamiento de la Informacin y Codificacin, Instituto de Física Aplicada, Consejo Superior de Investigaciones Científicas, Serrano 144, Madrid 28006, Spain
Abstract
We study in depth the equivalence between subshrubs and chaotic bands in the Mandelbrot set. In order to do so, we introduce the rules for chaotic bands and the rules for subshrubs, as well as the transformation rules that allow us to interchange them. From all the denominations of a chaotic band, we show the canonical form; that is, the one associated to the hyperbolic component that generates such a chaotic band. Starting from the study of the one-dimensional route, we fulfil an inductive study that gives a generalization of the shrub concept.