If you take a polynomial like ƒ(z) = z² − 1 and start to compose it with itself, you get higher and higher degree polynomials:

Discrete dynamics is the study of what happens when you take higher and higher compositions (or iterations) of ƒ; complex dynamics is discrete dynamics when the coefficients and the variable are all complex numbers. In general, there is no nice formula for what the n-th iterate of ƒ looks like. Fortunately, we can figure out a lot of useful properties of the iterates without actually finding formulas for them. In particular, we'll talk about periodic points (and especially attracting periodic points) and how they can be used to understand a lot about the dynamics of ƒ. That will lead us into a look at Fatou and Julia sets, and then to a brief introduction to the famous Mandelbrot set, which is more than just a pretty picture.

No knowledge of complex analysis will be assumed, but it will help to have at least seen complex numbers used before, even in passing.

USG funded