quadratic Julia set

For each complex number $c$, there is an associated quadratic map$f_c:ℂ\to ℂ$ 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 quadratic Julia set associated to$c$ and denote by $J_c$.

The function can also be viewed as having ${ℝ}^2$ as its domainand 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 thecollection of iterates $\{f^n:n\in \mathbb{N}\}$ is not a normalfamily can be roughly interpreted as saying that the Julia setincludes only those points exhibiting chaotic behavior. Inparticular, points whose orbit under $f$ goes to infinity are omittedfrom $J_c$, as well as points whose orbit converges to a point.

Sometimes for aesthetic purposes a Julia set is displayed with pointsof the latter type included. However, the chaoticity of the true Juliaset can be exploited to plot an approximation very quickly. Given asingle point $z$ known to be in the quadratic Julia set $J_c$, its inversesunder $f$, 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 onesquare root at each step) will be distributed fairly evenlyover $J_c$, so this gives a computationally inexpensive method to plotJulia sets.

Before the advent of computers, the French mathematician Gaston Juliaproved under what conditions a Julia set is connected or notconnected. After computers became available, it became possible tomake pictures displaying some of the complexity of these Julia sets,and the Mandelbrot set, a kind of index into connected quadratic Juliasets, 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 themhave been named accordingly. To give two examples: the San Marcodragon 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 andstill give very similar shapes).

Julia sets can be generalized to other iterated holomorphic functionson the complex plane or in a 3-dimensional space.