Discrete Dynamics in Nature and Society
Volume 2006 (2006), Article ID 70471, 25 pages

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

Received 17 January 2006; Accepted 13 March 2006

Copyright © 2006 G. Pastor et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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.