Algebraic Dynamics over Global Fields: Geometric and Analytic Methods

“该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。“这个研究项目的目的是研究全球和局部领域的代数动力学。 一些具体的目标详细介绍了这一建议是有关知名的开放问题的主题，包括Ih的猜想积分preperiodic点，一个功能领域模拟莫顿-西尔弗曼的统一有界猜想，并制定一个动态模拟的Szpiro猜想椭圆曲线。 所提出的技术包括使用Berkovich解析空间，规范高度函数，equidistribution，经典和非阿基米德Fatou-Julia理论，模空间的自同态，adelic容量理论，丢番图不等式。 从广义上讲，这个建议中的大多数思想可以被看作是两个一般主题的反映：第一个是曲线的数域和函数域之间的强平行，第二个是通过使用伯科维奇解析空间来统一处理阿基米德和非阿基米德代数动力学。代数动力学是一门结合了经典动力系统和代数几何的学科。 它的基本研究对象是一个集合X，代表某些代数方程的解，以及从X到自身的函数f。 说这些对象是在整体和局部域上定义的，实质上意味着它们对数论家和算术几何学家特别感兴趣。 当人们反复迭代函数f时，这个问题的动力学方面就开始发挥作用了。 例如，如果x是集合X的一个点，对于该点，无限序列f（x），f（f（x）），f（f（f（x），f（f（f（f（x），.，最终福尔斯落入一个有限循环，则x称为关于f的前周期点。 代数动力学中许多最基本的问题都涉及到前周期点的性质，但它们仍然没有得到很好的理解。 本研究计划旨在解决的自然开放性问题的一些例子包括：1。代数整数有多少个预周期点？ 2. 给定两个从X到自身的不同函数f和g，关于这两个函数有多少点是预周期的？3. 一个前周期点序列能以多快的速度接近一个给定的非前周期点？

"This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5)."The aim of this research project is to study algebraic dynamics over global and local fields. Some of the specific goals detailed in this proposal are related to well-known open problems in the subject, including Ih's conjecture on integral preperiodic points, a function field analogue of Morton-Silverman's uniform boundedness conjecture, and the formulation of a dynamical analogue of Szpiro's conjecture for elliptic curves. The proposed techniques include the use of Berkovich analytic spaces, canonical height functions, equidistribution, classical and non-archimedean Fatou-Julia theory, moduli spaces of endomorphisms, adelic capacity theory, and Diophantine inequalities. Broadly, the majority of the ideas in this proposal can be viewed as reflections of two general themes: the first is the strong parallel between number fields and function fields of curves, and the second is the unified treatment of archimedean and non-archimedean algebraic dynamics, through the use of Berkovich analytic spaces. Algebraic dynamics is a subject which combines elements of classical dynamical systems with algebraic geometry. Its basic object of study is a set X representing the solutions to certain algebraic equations, together with a function f from X into itself. To say that these objects are defined over global and local fields means essentially that they are of special interest to number theorists and arithmetic geometers. The dynamical aspect of the subject comes into play when one repeatedly iterates the function f. For example, if x is a point of the set X for which the infinite sequence f(x), f(f(x)), f(f(f(x))), f(f(f(f(x)))), ..., eventually falls into a finite loop, then x is called a preperiodic point with respect to f. Many of the most fundamental questions in algebraic dynamics involve the nature of preperiodic points, and yet they are still not very well understood. Some examples of natural open questions which this research proposal aims to address include the following: 1. How many preperiodic points are algebraic integers? 2. Given two different functions f and g from X into itself, how many points are preperiodic with respect to both functions? 3. How rapidly can a sequence of preperiodic points approach a given non-preperiodic point?