RUI: Arithmetic Dynamics: Algebraic and Analytic

Arithmetic dynamics is a mathematical field bridging the interface between number theory and dynamical systems. It concerns a broad range of algebraic questions about rational functions and polynomial equations that arise, along with the more analytic aspects of chaos and fractals, in the iteration of nonlinear functions. This project will focus on subtle open problems in this rich area, including the consideration of both the algebraic and the analytic aspects, along with the interactions between them. In addition, the PI will supervise two undergraduate students in a summer research REU experience, to expand their mathematical training. Any computational data and theoretical results from any part of this project will be shared via websites such as ArXiv.org, published in peer-reviewed mathematical journals, or otherwise disseminated openly to the broader mathematical research community. The dynamical systems studied here are defined by repeatedly composing a polynomial or rational function with itself. A wide range of chaotic properties arise under such iterations. The algebraic questions the PI will study in this project mainly concern the action of number-theoretic objects known as Galois groups on backwards orbits, which are natural dynamical objects. The analytic questions the PI will study concern the variation of a range of dynamical objects (especially the intricate fractals known as Julia sets) in a family of such dynamical systems, when working over a so-called non-archimedean field. These two sides of the project are tied together by p-adic dynamics, one of the PI's main areas of expertise; p-adic number fields comprise a fundamental example of non-archimedean fields. On the algebraic side, p-adic dynamical features for different prime numbers p can elucidate Galois actions. On the analytic side, the appropriate setting for p-adic and more general non-archimedean dynamics is the Berkovich projective line, another of the PI's areas of expertise. In both cases, the PI's application of p-adic dynamical tools promises to provide new insights into the unpredictable behavior of complicated arithmetic dynamical systems. 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.