Shapes of Julia sets, Thurston Sets, and Neural Networks

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Complex phenomena in a wide range of disciplines ranging from epidemiology to finance to climatology are modeled by dynamical systems. A dynamical system is a function from a space to itself. Points in the space represent possible states of the phenomenon and the function describes how the states evolve over time. Even relatively simple dynamical systems can exhibit very complicated long-term behaviors. Properties of the dynamical systems are often reflected in the shapes of associated mathematical sets. The principal investigator will investigate the shapes of dynamically defined sets that arise in three different contexts: (1) self-maps of intervals; (2) holomorphic dynamics, and (3) neural networks. In addition to advances in the theory of dynamical systems and geometry, results could lead to new techniques in computer graphics or machine learning. The three main goals of this project are to: (1) establish topological and geometrical properties of Thurston sets for various families of dynamical systems; (2) characterize the shapes of Julia sets of polynomials in one and several complex variables; (3) describe how network architecture constrains the decision regions of neural networks. The Thurston set for the family of superattracting unimodular self-maps of an interval is the closure of the set of all Galois conjugates of the exponentials of the topological entropies of all such maps. Plots of this set reveal that it has a rich and mysterious geometric structure. In previous work, the principal investigator characterized which subsets of the complex plane are approximable (in a strong sense) by Julia sets of polynomials. No analogous characterization of the possible global shapes of basins of attraction of infinity for polynomials in several complex variables is known, and the principal investigator will work to extend her techniques to this setting. Similarly, the principal investigator will investigate topological and geometrical obstructions to approximability by decision regions of neural networks of fixed network architectures.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.