RUI: Heights, Dynamics, and Preperiodic Points

DMS-0600878 Robert L. Benedetto这个项目涉及相对较新的数论动力学领域的几个问题。 在Morton和Silverman的一致有界性猜想中，一个中心问题是关于一个给定动力系统的前周期点的个数是否恰好是有理数。 虽然简单的提出，这个问题很快导致深刻和微妙的算术问题。 在过去十年左右的时间里，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，其中一些是由于研究人员。 特别是，最近几个进展，在动力学局部领域，在容量理论的填充Juliasets，并在研究所谓的当地的典型高度和动态绿色的功能，开辟了新的途径，为schoolingquestions的算术动力学。 由此产生的理论与复动力学的分析研究以及椭圆曲线和其他代数变体上有理点的算术研究都有相似之处;事实上，它已经引起了这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番图问题：计算自然产生的多项式方程组的有理数解集。 这些问题，再加上对素数的研究，推动了从丢番图和古希腊人到费马当的数论研究。 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也是开放的;该项目包括REU为本科生提供的暑期研究项目，以帮助他们进行数学训练。还计划进行密集的计算机计算，特别是正则高度和前周期点的计算，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.