External arguments of Douady cauliflowers in the Mandelbrot set

13 pages, 16 figures.-- Printed version published on Jun 2004.Near to the cusp of a cardioid of the Mandelbrot set, except for the main cardioid, there is a sequence of baby Mandelbrot sets. Each baby Mandelbrot set is in the center of a Douady cauliflower, a decoration constituted by an infinity of minute Mandelbrot sets and Misiurewicz points linked by filaments. A Douady cauliflower appears to have a complicated structure, and how the external rays land without intersecting in its Misiurewicz points and minute Mandelbrot sets is not really well known. In this work we study the Douady cauliflowers, giving a binary tree model to calculate the binary expansions of the external arguments of both the main cardioid of each minute Mandelbrot set and Misiurewicz points, and creating a horsetail model that explains the arrangement of its corresponding external rays.This work was supported by Ministerio de Ciencia y Tecnología of Spain, research grant TIC2001-0586.Peer reviewe