Abstract
We see unpredictable behavior around us every day: in the way weather patterns change, stock markets fluctuate, or wildfire spreads. We can also observe chaos in seemingly simple mathematical functions, and many recent undergraduate dynamical systems textbooks [5, 10, 11, 15] discuss chaotic systems. These texts all include chapters covering Julia sets, the part of the domain where a complex function behaves chaotically. A Julia set is usually an intricate and beautiful object, and images of Julia sets can be found on posters, book covers, T-shirts, screen savers, and web pages. Many books [10, 11, 15] focus on Julia sets of complex polynomials of degree 2 or 3, and even the casual reader will observe from the pictures that all of these Julia sets appear to have area zero. For polynomials, it turns out that there must always be some large region where the function has very predictable behavior. What if we examine other types of functions? Is there a complex function whose Julia set is everything? That is, does there exist a complex function that acts chaotically on its entire domain? The picture of such an example would be entirely black.
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