RUI: Heights, Dynamics, and Preperiodic Points

本项目涉及数论动力学这一相对较新的领域的几个问题。在Morton和Silverman的一致有界性猜想中，一个中心问题是关于给定动力系统中恰好是有理数的周期前点的数目。这个问题虽然简单，但很快就会引出深奥而微妙的算术问题。在过去的十多年里，在这些问题上已经取得了很大的进展，并且已经开发了大量的技术机器来帮助回答这些问题，其中一些要归功于研究者。特别是，最近在局部场动力学、填充Juliasets的容量理论、所谓的局部正则高度和动力学格林函数的研究方面的一些进展，为考虑算术动力学问题开辟了新的途径。所得到的理论与复杂动力学的解析研究和椭圆曲线上有理点的算术研究及其他代数变种都有相似之处；事实上，它已经引起了两个领域专家的兴趣。最终，该项目的目标是研究一类经典丢芬图问题：计算自然产生的多项式方程集的有理数解集。这些问题，连同对质数的研究，推动了数论的研究，从丢番图和古希腊人到费马多一直到今天。除了这些问题的内在美之外，它们还在编码理论和密码学中得到了应用。此外，算术动力学的某些方面对本科生来说是容易理解的；该项目包括REU为本科生提供的夏季研究项目，以帮助他们进行数学训练。还计划进行密集的计算机计算，特别是规范高度和周期前点的计算，并将生成的任何相关数据发布或张贴在网站上，以造福于更大的研究社区。

DMS-0600878Robert L. BenedettoThis project concerns several problems from the relatively new fieldof number theoretic dynamics. One central question, formalized in theUniform Boundedness Conjecture of Morton and Silverman, asks about thenumber of preperiodic points of a given dynamical system that happento be rational numbers. While simple to pose, the question quicklyleads to deep and subtle arithmetic problems. Over the past decade orso, there has been great progress on such questions, and a substantialamount of technical machinery, some due to the investigator, has beendeveloped to help answer them. In particular, several recent advancesin dynamics over local fields, in the capacity theory of filled Juliasets, and in the study of so-called local canonical heights anddynamical Green's functions, have opened new avenues for consideringquestions of arithmetic dynamics. The resulting theory has parallelsboth to the analytic study of complex dynamics and to the arithmeticstudy of rational points on elliptic curves and other algebraicvarieties; indeed, it has drawn interest from specialists in bothfields.Ultimately, the goal of the project is to study a type of classicalDiophantine problem: the computation of the set of rational numbersolutions to a naturally arising set of polynomial equations. Suchproblems, together with the study of primes, have driven the study ofnumber theory from Diophantus and the ancient Greeks through Fermatand into the present day. Besides the intrinsic beauty of suchproblems, they have since found applications in coding theory andcryptography. In addition, certain aspects arithmetic dynamics areaccessible to undergraduates; the project includes REU summer researchprojects for undergraduates to aid in their mathematical training.Intense computer computations, especially of canonical heights andpreperiodic points, are also planned, with any relevant data generatedto be published or posted on a website, for the benefit of the largerresearch community.