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.BENEDETTOTHIS项目涉及相对较新的Fieldof Number理论动力学的几个问题。 一个中心问题是在莫顿和西尔弗曼的统一界面构想中正式化的，询问了给定动力学系统的预碘点的含量，这是理性数字。 虽然简单地摆姿势，但这个问题迅速涉及深度和微妙的算术问题。 在过去的十年中，在此类问题上取得了长足的进步，并且大量的技术机械（有些由于研究人员而导致的技术机械）已努力帮助回答他们。 特别是，在填充的juliasets的能力理论以及对所谓的本地规范高度和山格林的功能的研究中，最近的几种前进动力学为考虑算术动力学的启动开辟了新的途径。 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方面的理性点的算术研究具有相似之处。的确，它引起了双场专家的兴趣。最重要的是，该项目的目的是研究一种古典饮食问题：对自然产生的一组多项式方程组的一组合理数字的计算。 这样的问题以及对素数的研究，将研究的研究驱动了从毒蛇和古希腊人的研究，直到今天，就通过了Fermatand。 除了这种问题的内在美之外，他们还发现了编码理论和屏幕术中的应用。 此外，某些方面的算术动力学可以使本科生可以探索；该项目包括针对本科生的REU夏季研究项目，以帮助他们进行数学培训。还计划了计算机计算，特别是规范高度和Preperiodic观点，并在网站上发布或在网站上发布任何相关数据，以供大型研究社区发布或发布。