What is Complex Dynamics?

This webpage gives a quick rundown of the topics and character that one might expect when taking Mark's Math 393 special topics class on Complex Dynamics.

Culturally

The topics of complex dynamics include Julia sets, the Mandelbrot set, and the fabulous images of these things. If you're curious about how these kinds of pictures can arise from pure mathematics, then this class is for you!

Mathematically

The term "Complex Dynamics" is defined by two words: complex and dynamics. Let's dissect these two, focussing on the second one first.

Discrete Dynamics

In pure mathematics, we often think of in terms of iteration. That is, we have a function $f$ that maps some set to itself and we iterate that function. Thus, we pick an initial value $x_1$, we plug that value back into $f$ to get $x_2$, we plug that value back into $f$ to get $x_3$, etc. We then study the types of sequences that can result.

For example, if $f(x)=x^2$ and $x_1=1/2$, then $$\begin{align} x_2 &= (1/2)^2 = 1/4 \\ x_3 &= (1/4)^2 = 1/16 \\ x_4 &= (1/16)^2 = 1/256 \\ &\ldots \text{etc} \end{align}$$ Note that we generate a sequence of numbers called the orbit of the point under iteration of the function. In this case, it looks like we generated an orbit that tends quickly to zero. In fact, if we start with any number $x$ with $|x|<1$, we should again generate a sequence that tends quickly to zero.

Complex variables

The term complex arises from the fact that the domain of the function that we iterate is the set of complex numbers. That is, we use numbers of the form $z=a+bi$ where $i^2=-1$.

These kinds of numbers can be visualized in a coordinate plane, like the the image below. The axes are called the real and imaginary axes. Absolute value, in this context, represents the distance to the origin. Thus, the circle you see is $$\{z\in\mathbb C:|z|=1\},$$ i.e. the set of all complex numbers whose absolute value is equal to 1.

Complex iteration

In complex dynamics, we iterate a function of a complex variable. The picture above is meant to illustrate the orbit of the function $f(z)=z^2$. If you are looking at this on a computer, you should be able to hover your mouse over the picture to pick an initial point and see the resulting orbit drawn on top of the image. Note that all points inside the circle lead to orbits that converge to zero while all points outside the circle lead to orbits that diverge to infinity. The dynamics right on the circle are quite sensitive to the initial point - a hallmark of chaos.

Other functions that are even slightly more complicated can lead to other types of behavior. The picture below, for example, illustrates the possibilities for $$f(z) = z^2 - 0.123 + 0.745 i.$$

Prequisites and reading

• The technical prereqisite for this course is Calculus II. You should be comfortable with the idea of a sequence and not afraid to think about them geometrically in the complex plane.
• We will play with the computer a bit using tools like the following:
  • Julia sets and the Mandelbrot set
  • Polynomial Julia sets
  • Rational Julia sets
  • Newton's method for complex polynomials
• Here's the first draft of text on complex dynamics. I'll probably add quite a bit of intro material to it, though.
• You can learn the basics of complex numbers in this text. We really only need a bit of it, we'll cover chapter 1 and section 3.1 in the course.