Complex Algebraic Dynamics and Geometry

Northwestern University, Evanston IL

Investigators

Abstract

The repeated application of a function produces sequences of values whose behavior can be quite surprising. Such discrete dynamical systems occur as models throughout science and engineering. Polynomials and rational functions of a single variable provide basic examples of non-invertible, algebraic, discrete dynamical systems by iteration. Even the simplest families of examples exhibit complicated and chaotic dynamical behavior; the most famous is the family of quadratic polynomials where a point x is mapped to x "squared" plus a constant. The constant can be a real or complex number. This family gives rise to the Mandelbrot set, which continues to baffle researchers. A fundamental problem in the mathematical study of these systems is to characterize their stability: under what circumstances -- and by how much -- can we perturb a system while maintaining its long-term dynamical features? The primary goal of this research project is to explore connections between the stability of algebraic dynamical systems and the algebraic or geometric structures that are preserved under iteration. The first connections between the algebra and dynamics of these models were discovered in the 19th century. This project will exploit modern tools from arithmetic geometry and dynamical systems to strengthen these connections and deepen our understanding of the systems themselves. The project studies: (1) the arithmetic properties of elliptic curves and abelian varieties, with dynamical methods; (2) a series of substantial conjectures about "unlikely intersections" in moduli spaces of dynamical systems; and (3) the 3-dimensional Euclidean geometry of a rational function, with curvature form equal to its measure of maximal entropy. The main goal of this project is to build relations between the algebra and the geometry of dynamical systems on algebraic varieties, with an eye towards applications in Diophantine geometry. The second goal of the project is a study of rational maps in dimension one, particularly an exploration of the canonical shape of the dynamical system and its stability properties. In recent work, the investigator and collaborator developed new methods of proof incorporating tools from both complex and non-Archimedean analysis and formulated a conjecture about the dynamics of rational maps that encompasses known results about elliptic curves. This project aims to investigate particular cases of this conjecture while developing the theory to connect these results to the study of the symmetries and stability and invariants of complex dynamical systems in dimensions one and two.