# Dynamics on the complex plane

# Dynamics on the complex plane

Iterations of an analytic function of a complex variable is a wonderful laboratory for chaos. After classifying Mobius transformations by the nature of their fixed points, this chapter turns to iterations of rational functions. Mathematical notions include the Riemann sphere, metric spaces, and compactness. The Julia set of a map is the closure of its unstable periodic point. In essence, that is where the map is chaotic. The Fatou set is its complement. Some examples of Julia sets are plotted, as is the Mandelbrot set.

*Keywords:*
chaos, Mobius transformations, unstable periodic point, compactness, Julia set, Fatou set

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