Julia sets

Remember the squaring function

Our groovy complex squaring demo illustrates some pretty interesting behavior. Specifically:

• There are two stable regions: the interior and the exterior of the unit circle.
• The dynamics on the boundary of the two stable regions are chaotic.

Julia sets

Very similar behavior happens when we iterate pretty much any polynomial. Let's generalize just a bit to polynomials of the form

You can play with this family right here:

The image on the left is called the Mandelbrot set. It lives in the complex plane. When you click on it, you have essentially chosen a point \(c\) and defined the function \(f_c\). As a result, an image appears on the right; this is called the Julia set or filled Julia set of \(f_c\).

The escaping orbit

Lemma: Suppose that \(|z_0|\geq|c|\) and that \(|z_0|>2\). Then, \(|f_c(z_0)| > |z_0|\) and the orbit of \(z_0\) under iteration of \(f_c\) will diverge to \(\infty\).

Proof: Using the triangle inequality, we have: \[\begin{align} |z_0^2| &= |z_0^2+c-c| \\\ &\leq |z_0^2+c|+|-c| \\\ &= |z_0^2+c| + |c|. \end{align}\] So that \[\begin{align} |z_0^2+c| &\geq |z_0^2|-|c| \\\ &\geq |z_0^2|-|z_0| \\\ &=|z_0|(|z_0|-1)>|z_0|. \end{align}\] In fact, if we let \(\lambda = |z_0|-1\) and define \(z_{i+1}=f_c(z_i)\), then \(|f_c(z_{i+1})| \geq \lambda f_c(z_i)\) for all \(i\). That is, each term in the orbit is larger than the previous term by at least this same common factor. As a result, the orbit diverges to \(\infty\).

Corrollary: Suppose we iterate \(f_c\) from some initial seed \(z_0\in\mathbb C\) and find that, for some \(k\in\mathbb N\), \(|f_c^k(z_0)|>\max(c,2)\). Then \[\lim_{n\to\infty} f_c^n(z_0) = \infty.\] In other words, if some iterate ever exceeds both \(c\) and \(2\) in absolute value, then the orbit escapes.

The Julia set

Definition: The filled Julia set of a polynomial \(f\) is exactly the set of all initial seeds \(z_0\in\mathbb C\) that lead to bounded orbits under iteration of \(f\). The Julia set of \(f\) is the boundary of the filled Julia set.

Since the Julia set is the boundary of the filled Julia set any neighborhood (no matter how small) of any point in the Julia set contains points that stay bounded under iteration of \(f_c\) and points that diverge to \(\infty\) under iteration of \(f_c\). The Julia set is exactly where the chaos lies under iteration of \(f_c\).

The escape time algorithm

We can generate an image of the filled Julia of the quadratic polynomial \(f_c(z)= z^2+c\) as follows.

• Choose an escape radius. Our computations indicate that the maximum of \(|c|\) and \(2\) should do.
• Choose a maximum possible number of iterations. The correct choice really depends on the function and how close you want to zoom in. For a global view of a Julia set of a function of the form \(f_c(z)=z^2+c\), a maximum iteration count of \(100\) is a sufficient and popular choice. Generally, the higher the maximum iteration count, the more accurate the picture but the longer the computation time.
• Choose a grid of initial seeds in the complex plane. These will be represented as pixels on our computer screen and we'll generate our image by shading each of these points.
• For each point in your grid of initial seeds, iterate the function \(f\) from that point until one of two things happens.
  • The iterate exceeds our escape radius, in which case we shade the pixel according to how many iterates it took to get there, or
  • We reach our maximum allowable number of iterates, in which case we color the pixel black.