SetDeJulia 2007-06-12 17:33:08 2007-06-16 10:49:20 Definition Julia set % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here For each complex number $c$, there is an associated quadratic map $f_c\colon\mathbb{C}\to\mathbb{C}$ defined by $f_c(z) = z^2 + c$. Since polynomials are analytic, it follows that $f_c$ has a Julia set $J(f_c)$, which we call the \emph{quadratic Julia set} associated to $c$ and denote by $J_c$. The function can also be viewed as having $\mathbb{R}^2$ as its domain and codomain. If $c = a + ib$, then \[ f_c(x, y) = (x^2 - y^2 + a, 2xy + b). \] The characterization of the Julia set $J_c$ as all points $z$ for which the collection of iterates $\{f^n\colon n\in\mathbb{N}\}$ is not a normal family can be roughly interpreted as saying that the Julia set includes only those points exhibiting chaotic behavior. In particular, points whose orbit under $f$ goes to infinity are omitted from $J_c$, as well as points whose orbit converges to a point. Sometimes for aesthetic purposes a Julia set is displayed with points of the latter type included. However, the chaoticity of the true Julia set can be exploited to plot an approximation very quickly. Given a single point $z$ known to be in the quadratic Julia set $J_c$, its inverses under $f$, that is, the square roots of $z - c$, are also in $J_c$. Moreover, by the chaoticity condition the ``backwards orbit'' of $z$ (selecting just one square root at each step) will be distributed fairly evenly over $J_c$, so this gives a computationally inexpensive method to plot Julia sets. Before the advent of computers, the French mathematician Gaston Julia proved under what conditions a Julia set is connected or not connected. After computers became available, it became possible to make pictures displaying some of the complexity of these Julia sets, and the Mandelbrot set, a kind of index into connected quadratic Julia sets, was discovered. In the same way that some people see recognizable shapes in clouds, some people see recognizable shapes in Julia sets, and some of them have been named accordingly. To give two examples: the San Marco dragon at $\frac{-3}{4} + 0i$ and the Douady rabbit at $\frac{-1}{8} + \frac{745}{1000}i$ (the coordinates can be varied by small values and still give very similar shapes). % not sure what to do with this sentence Julia sets can be generalized to other iterated holomorphic functions on the complex plane or in a 3-dimensional space. \begin{thebibliography}{1} \bibitem{hl} H. Lauwerier, translated by Sophia Gill-Hoffst\"adt. {\it Fractals: Endlessly Repeated Geometric Figures} Princeton: Princeton University Press (1991): 124 - 151 \end{thebibliography} \PMlinkescapeword{index} \PMlinkescapeword{similar} \PMlinkescapeword{type}