RUI: Heights, Dynamics, and Preperiodic Points

DMS-0600878罗伯特·L·贝内德托这个项目涉及数论动力学这个相对较新的领域的几个问题。在Morton和Silverman的Uniform有界性猜想中形式化的一个中心问题是关于给定动力系统中碰巧是有理数的准周期点的数目。虽然这个问题很简单，但很快就会引出深刻而微妙的算术问题。在过去的十多年里，在这些问题上取得了很大的进展，已经开发了大量的技术机器来帮助回答这些问题，其中一些是由调查人员提供的。特别值得一提的是，在局域上的动力学、充满Julias集的容量理论、以及在所谓的局域正则高度和动力格林函数的研究中，最近的几个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和椭圆曲线上有理点的算术研究以及其他代数变体具有相似之处；事实上，它引起了这两个领域的专家的兴趣。最终，该项目的目标是研究一类经典的丢番图问题：自然产生的一组多项式方程的有理数解的计算。这些问题和对素数的研究一起推动了数论的研究，从丢番图到古希腊人通过费马特一直到今天。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，本科生可以接触到算术动力学的某些方面；该项目包括为本科生提供的REU暑期研究项目，以帮助他们的数学培训。还计划进行密集的计算机计算，特别是典型高度和周期前点的计算机计算，产生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。

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.