Directions in arithmetic dynamics

算术动力学方向

• 批准号: 1200749
• 负责人: Thomas Tucker
• 金额: $ 12万
• 依托单位: University of Rochester
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-08-01 至 2015-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1200749&HistoricalAwards=false
• 关键词: Directions; arithmetic; dynamics

This project focuses on the subject of algebraic dynamics. Orbits of points under iterates of maps are among the most important objects in dynamics. In the study of algebraic dynamics, the map is typically a polynomial or rational map in one or more variables. One natural question is what kinds of algebraic relations the forward orbits of points can satisfy. When the orbits are infinite (corresponding to so-called "wandering points"), one natural question corresponds to the well-known Mordell-Lang conjecture for abelian varieties. When the orbits are finite (corresponding to "preperiodic points'), a natural question is an analog of the Manin-Mumford-Bogomolov conjecture for abelian varieties. The primary purpose of this project is to advance knowledge about these two questions, via a combination of techniques from p-adic analysis, geometry, and diophantine approximation.This project focuses on the interaction between algebraic maps and algebraic equations. An algebraic map is a function such as f(x) = 2x+3. Applying the map repeatedly to a single number gives what is called the orbit of that number under f. For example, if f(x) = 2x+3 and we start with the number 1, then the orbit is 1, 5, 13, 29, 61, and so on. One can also form orbits out of pairs, triplets, and n-tuples of numbers by allowing f to be an algebraic map in more than one variable. An algebraic equation is simply a more general version of the familiar quadratic polynomials one encounters in beginning algebra, except that it may have more variables and have higher degree terms. These orbits exhibit many interesting properties. On the one hand they have so little structure that they may be good engines for the generation of random numbers; on the other, one might hope that their algebraic and arithmetic properties may give rise to new analogs the theories of fractals and chaos, which in many cases arose from the consideration of geometric and analytical properties of orbits.

这个项目的重点是代数动力学的主题。 点在映射迭代下的轨道是动力学中最重要的对象之一。 在代数动力学的研究中，映射通常是一个或多个变量的多项式或有理映射。 一个自然的问题是点的前向轨道能满足什么样的代数关系。 当轨道是无限的（对应于所谓的“游荡点”），一个自然的问题对应于著名的莫德尔-朗猜想阿贝尔品种。 当轨道是有限的（对应于“前周期点”），一个自然的问题是一个类似的Manin-Mumford-Bogomolov猜想的阿贝尔品种。 本项目的主要目的是通过结合p-adic分析、几何学和丢番图近似的技术来提高对这两个问题的认识。本项目的重点是代数地图和代数方程之间的相互作用。代数映射是一个函数，如f（x）= 2x+3。 将该映射重复应用于单个数，就会给出该数在f下的所谓轨道。 例如，如果f（x）= 2x+3，我们从数字1开始，那么轨道是1，5，13，29，61，等等。我们也可以通过允许f是一个多变量的代数映射来形成由数对，三元组和n元组组成的轨道。 一个代数方程只是一个更一般的版本熟悉的二次多项式在开始代数，除了它可能有更多的变量和更高的次数。 这些轨道表现出许多有趣的性质。 一方面，它们的结构如此之少，以至于它们可能是生成随机数的良好引擎;另一方面，人们可能希望它们的代数和算术性质可能会产生新的类似物--分形和混沌理论，在许多情况下，这些理论是从考虑轨道的几何和分析性质中产生的。