This project concerns the development of analytic tools on Berkovich spaces, with applications to the dynamics of rational functions. Three problems will be investigated. First, the Principal Investigator will address a finiteness conjecture of Ih, which asserts that for any dynamical system associated to a rational function over a number field, there are only finitely many preperiodic points which are integral with respect to a fixed non-preperiodic point. The attack involves proving an analogue, for preperiodic points, of Baker's theorem on linear forms in logarithms. Second, using a recently-developed Arakelov theory for the Berkovich projective line, P.I. will study the no-wandering domains conjecture for rational maps over nonarchimedean local fields. Third, the P.I. will investigate pluripotential theory on Berkovich spaces of arbitrary dimension, with the goal of defining a nonarchimedean Monge-Ampere operator and using it to create an adelic arithmetic intersection theory. In these investigations, the P.I. will collaborate with other members of an emerging group in arithmetic dynamics. He will also mentor University of Georgia graduate students and recent number theory Ph.D.'s in the southeastern United States.
This project involves research at the interface of two branches of mathematics, analysis (the study of functions) and dynamics (the study of iteration), with a focus on questions related to number theory. One of the scientific goals of the project is to prove a general finiteness conjecture for dynamical systems attached to rational functions. Two important cases of the conjecture, for roots of unity and for torsion points on elliptic curves, were proved under the P.I.'s previous NSF grant. The planned attack involves generalizing a fundamental arithmetical tool used in the earlier proof, to the context of dynamical systems. A second goal is to complete the classification of dynamical systems for rational functions defined over p-adic fields, showing that an analogue of the well-known "no wandering domains" theorem of Sullivan, proved for rational functions over the complex numbers, holds in the nonarchimedean locally compact case as well. A third goal is to extend a theory of subharmonic functions on the p-adic projective line, developed by the P.I., to higher-dimensional p-adic varieties. This will be used to construct an Arakelov theory for higher-dimensional varieties which treats archimedean and nonarchimedean contributions in a parallel, analytical way. In carrying out this research, the P.I. plans to collaborate extensively with other mathematicians, both junior and senior; one of the infrastructure goals of the project is to strengthen an emerging interdisciplinary research group focusing on number-theoretic aspects of dynamics. Another infrastructure goal is to mentor current University of Georgia doctoral students, and to main ties with and encourage graduates of the UGA number theory program who are now teaching at small colleges in the southeastern United States.
