RUI: Heights, Dynamics, and Preperiodic Points

RUI：高度、动态和前期点

• 批准号: 0600878
• 负责人: Robert Benedetto
• 金额: $ 11.3万
• 依托单位: Amherst College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-07-01 至 2009-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0600878&HistoricalAwards=false
• 关键词: RUI; Heights; Dynamics; Preperiodic; Points

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.

DMS-0600878Robert L. Benedetto 该项目涉及相对较新的数论动力学领域的几个问题。 莫顿和西尔弗曼的一致有界猜想形式化的一个核心问题是询问给定动力系统恰好是有理数的前周期点的数量。 虽然提出起来很简单，但这个问题很快就会导致深刻而微妙的算术问题。 在过去的十年左右的时间里，这些问题已经取得了很大的进展，并且已经开发出了大量的技术机制来帮助回答这些问题，其中一些技术机制是由研究者开发的。 特别是，局部域动力学、填充 Julia 集的容量理论以及所谓局部正则高度和动态格林函数的研究方面的一些最新进展，为考虑算术动力学问题开辟了新的途径。 由此产生的理论与复杂动力学的分析研究以及椭圆曲线和其他代数簇上有理点的算术研究相似。事实上，它引起了两个领域专家的兴趣。最终，该项目的目标是研究一种经典的丢番图问题：计算一组自然产生的多项式方程组的有理数解集。 这些问题与素数的研究一起，推动了从丢番图和古希腊人到费马丹的数论研究一直到今天。 除了此类问题的内在美之外，它们还在编码理论和密码学中找到了应用。 此外，算术动力学的某些方面可供本科生学习；该项目包括为本科生提供的 REU 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，生成的任何相关数据都将发布或发布在网站上，以造福于更大的研究社区。