George Pólya Awards for 2022

Abstract

In “The Beautiful Chaotic Dynamics f i,” Joseph and Michelle Previte guide their readers on an engaging exploration of the principal branch of the complex map f (z) = i. While Brouwer’s Fixed-Point Theorem guarantees that this function has at least one fixed point, the authors establish that there are, in fact, an infinite number of fixed points—all but one of which are unstable. Of course, with this initial groundwork in place, exotic and ever-enchanting fractal images cannot be far behind! By iterating i numerically for a large collection of initial points, the Prevites create a graph to identify those points in the plane which lie in the basin of attraction of the stable fixed point and those initial points which escape to infinite. Earlier in the paper, technology was used to explore the locations of the sought after fixed points and to follow up with careful mathematical analysis to verify the information alluded to in the resulting graphs. This helpful side of technology is counter-balanced as the Prevites use mathematical analysis to carefully point out the limits of technology by identifying points within the basin of attraction that the computer-generated plot clearly mis-identified. While some of the points the computer identified as being outside the basin of attraction actually approach the stable fixed point, the authors provide a compactness argument to show that there are indeed points in the plane with orbits whose moduli tend to infinity. The paper continues by examining the composite maps f 2(z) and f 3(z) to identify period two and period three points of f (z). Thus, one concludes that f (z) is a chaotic map having periods of all orders. The authors conclude by giving readers six open problems to investigate on their own. The Prevites’ clear exposition makes it easy for a reader to interact with this paper at a variety of levels. There are five exercises sprinkled throughout the paper that allows one to, at first, skip some of the technical details and more quickly get to the “good stuff”—that is, the beautiful chaotic dynamics of i. However, these exercises contain some very nice analysis for students to grapple with and help to reiterate the usefulness of one-sided limits, monotonicity, and notions of convergence that students have likely seen in their mathematics courses. Students who already have some familiarity with complex numbers could use this paper as a nice introduction to the ideas of fractals and chaos. Taking time to fill in some of the details, to reproduce some of the lovely plots, and to explore the open problems would make for a truly engaging and worthwhile project for students and instructors alike.