Analysis on Berkovich spaces and Arithmetic dynamics

DMS-0601037罗伯特·鲁姆利这个项目涉及Berkovich空间上分析工具的开发，以及有理函数动力学的应用。将调查三个问题。首先，首席调查者将解决ih的一个有限猜想，该猜想断言对于与数域上的有理函数相关联的任何动力系统，只有有限多个预周期点相对于固定的非预周期点是积分的。这一攻击涉及证明一个类似于贝克关于对数形式的线性形式的对数周期前点的定理。其次，利用最近发展的关于Berkovich投影线的Arakelov理论，P.I.将研究非阿基米德局部域上有理映射的非游荡域猜想。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Monge-Ampere算子，并用它来创建一个非阿德勒算术交集理论。在这些调查中，私家侦探将与一个新兴的算术动力学小组的其他成员合作。他还将在美国东南部指导佐治亚大学的研究生和最近获得数论博士学位的S。这个项目涉及两个数学分支的界面研究，分析(函数研究)和动力学(迭代研究)，重点是与数论相关的问题。该项目的科学目标之一是证明与有理函数有关的动力系统的一般有限性猜想。这一猜想的两个重要例子--单位根和椭圆曲线上的扭点--在P.I.S以前的国家自然科学基金资助下得到了证明。计划中的攻击包括将早期证明中使用的基本算术工具推广到动力系统的背景下。第二个目标是完成定义在p-adad域上的有理函数动力系统的分类，证明了对于复数上的有理函数，著名的Sullivan的“无游荡域”定理的类似证明在非阿基米德局部紧的情况下也成立。第三个目标是将P.I.发展的p-进射线上的次调和函数理论推广到更高维的p-进射域。这将被用来构建高维变种的阿拉克洛夫理论，该理论以平行的、分析的方式对待阿基米德和非阿基米德的贡献。在开展这项研究时，P.I.计划与其他初级和高级数学家广泛合作；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。另一个基础设施目标是指导佐治亚大学目前的博士生，并与目前在美国东南部小型大学任教的UGA数论项目的毕业生建立主要联系并鼓励他们。

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.