RUI: Boundedness questions in arithmetic dynamics

“这个奖项是根据2009年美国复苏和再投资法案(公法111-5)资助的。”这个项目涉及近年来在数论动力学领域出现的两个关键猜想的主题。第一个猜想由Morton和Silverman在1994年提出，它预言了一个给定动力系统中恰好是有理数的预周期点的个数存在一个一致的上界；第二个猜想指出一个非准周期有理点的正则高度存在一个一致的下界；换句话说，粗略地说，这样一个点的非预周期性必然在动力系统有限次迭代后变得明显。虽然这两种猜想都很容易通过例子或正式陈述来描述，但它们很快就会导致深刻而微妙的算术问题。在过去的十年或二十年里，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械来帮助回答这些问题：局部场动力学的进展、Julia集的容量理论、局部正则高度和动态格林函数。由此产生的理论与复杂动力学的分析研究以及椭圆曲线上有理点的算术研究和其他代数变体具有相似之处，并且引起了这两个领域的专家的兴趣。这个项目的核心是一类经典的丢番图问题：理解自然产生的多项式方程组的有理数解的集合。几千年来，这些问题一直是数论研究的一个主要主题。更具体地说，这个项目的一个中心目标是加深我们对由某种动力学引起的丢番图问题的理论理解：多项式或有理函数的迭代。同时，由于数论的这个动态子领域相对年轻，它的许多方面对本科生来说是可以理解的，但仍然没有解决的问题。因此，就像在早期成功的项目中一样，调查人员计划指导一些REU暑期研究项目的学生，以帮助他们进行数学训练。REU将包括密集的计算机计算，特别是典型高度和前周期点的计算机计算，产生的任何相关数据将通过出版物和网络传播给研究界。

"This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5)."This project concerns topics surrounding two key conjectures that have arisen in recent years in the field of number-theoretic dynamics. The first conjecture, stated by Morton and Silverman in 1994, predicts that there is some uniform upper bound for the number of preperiodic points of a given dynamical system that happen to be rational numbers.The second conjecture states that there is a uniform lower bound for the canonical height of a non-preperiodic rational point; in other words, roughly speaking, the non-preperiodicity of such a point must become obvious after a bounded number of iterations of the dynamical system. While easy to describe either by example or by formal statement, both conjectures quickly lead to deep and subtle arithmetic problems. Over the past decade or two, there has been great progress on such problems, and a substantial amount of technical machinery, some due to the investigator, has been developed to help answer them:advances in dynamics over local fields, capacity theory of Julia sets, local canonical heights, and dynamical Green's functions. The resulting theory has parallels both to the analytic study of complex dynamics and to the arithmetic study of rational points on elliptic curves and other algebraic varieties, and it has drawn interest from specialists in both fields.At its heart, the focus of this project is a type of classical Diophantine problem: to understand the set of rational number solutions to a naturally arising set of polynomial equations. Such problems have been a major theme in the study of number theory for thousands of years. More specifically, a central goal of this project is to further our theoretical understanding of Diophantine problems arising from a certain kind of dynamics: the iteration of a polynomial or rational function. At the same time, because this dynamical subfield of number theory is relatively young, many of its aspects are accessible to undergraduates but still remain unsolved problems.Thus, as in an earlier successful project, the investigator plans to supervise some students in an REU summer research project to aid in their mathematical training. The REU will include intense computer computations, especially of canonical heights and preperiodic points, and any relevant data generated will be disseminated to the research community via publications and the web.