Discrete Dynamics in Nature and Society
Volume 2011 (2011), Article ID 837262, 23 pages
http://dx.doi.org/10.1155/2011/837262
Research Article

Calculation of the Structure of a Shrub in the Mandelbrot Set

G. Pastor,1 M. Romera,1 A. B. Orue,1 A. Martín,1 M-F. Danca,2 and F. Montoya1

1Instituto de Física Aplicada, CSIC, Serrano 144, 28006 Madrid, Spain
2Romanian Institute of Science and Technology, Cireşilor 29, 400487 Cluj-Napoca, Romania

Received 26 October 2010; Revised 19 January 2011; Accepted 1 March 2011

Academic Editor: Recai Kilic

Copyright © 2011 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.

Abstract

We calculate the external arguments of the structure of any shrub in the Mandelbrot set. Before calculating, we revise, expand, and clarify some tools useful for this paper: harmonics, pseudoharmonics, the concept of structure, the structure of a shrub, and the ancestral route. Finally we present the main contribution of this paper, a three-step algorithm which allows us to calculate the structure of the shrub. In the first step, we use pseudoharmonics that were previously introduced by us, in order to calculate the first and last external arguments of a structural node. In the second step, starting from two general properties of the Misiurewicz points external arguments introduced here by us, we present a new method to calculate the intermediate external arguments. In the last step we introduce a third property that allows us to calculate the external arguments of the representatives of the branches emerging from the structural nodes.