Analysis on Berkovich spaces and Arithmetic dynamics

这个项目关注Berkovich空间上的分析工具的发展，并将其应用于有理函数的动力学。将调查三个问题。首先，研究人员将讨论Ih的有限性猜想，该猜想断言，对于与数域上的有理函数相关的任何动力系统，只有有限多个预周期点对一个固定的非预周期点积分。这种攻击涉及到在对数的线性形式上证明一个类似于贝克定理的周期前点。其次，使用最近发展的Arakelov理论的Berkovich投影线，P.I.将研究非阿基米德局部场上的有理地图的无游荡域猜想。第三，P.I.将研究任意维Berkovich空间上的多势理论，目的是定义一个非阿基米德蒙日-安培算子，并用它来创建一个阿基米德算术相交理论。在这些调查中，私家侦探将与算术动力学新兴小组的其他成员合作。他还将指导佐治亚大学的研究生和美国东南部最近的数论博士学位。这个项目涉及数学的两个分支，分析（函数的研究）和动力学（迭代的研究）的接口研究，重点是与数论相关的问题。该项目的科学目标之一是证明附属于有理函数的动力系统的一般有限性猜想。在P.I.下，证明了该猜想的两个重要情况，即关于单位根和关于椭圆曲线上的扭转点这是他之前获得的NSF资助。计划中的攻击涉及将先前证明中使用的基本算术工具推广到动力系统的背景下。第二个目标是完成在p进域上定义的有理函数的动力系统的分类，并证明对于复数上的有理函数证明的著名的沙利文“无漫游域”定理的类似物，在非阿基米德局部紧化情况下也成立。第三个目标是将pi发展的p进射影线上的次调和函数理论推广到更高维度的p进变换。这将用于构建更高维度的Arakelov理论，该理论以平行的分析方式处理阿基米德和非阿基米德的贡献。在进行这项研究时，P.I.计划与其他数学家广泛合作，包括初级和高级数学家；该项目的基础设施目标之一是加强一个新兴的跨学科研究小组，专注于动力学的数论方面。另一个基础设施的目标是指导当前佐治亚大学的博士生，并与佐治亚大学数论课程的毕业生建立主要联系，鼓励他们现在在美国东南部的小型学院任教。

DMS-0601037Robert Rumely 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.