Directions in arithmetic dynamics

这个项目的重点是代数动力学。地图迭代下的点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映射或有理映射。一个自然的问题是，点的正向轨道可以满足什么样的代数关系。当轨道是无限的（对应于所谓的“流浪点”），一个自然的问题对应于著名的莫德尔-朗猜想的阿贝尔变种。当轨道是有限的（对应于“周期前点”），一个自然的问题是一个类似于阿贝尔变的Manin-Mumford-Bogomolov猜想的问题。这个项目的主要目的是通过p进分析、几何和丢番图近似技术的结合，提高对这两个问题的认识。这个项目的重点是代数映射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。将这张图反复应用到一个数字上，就会得到这个数字在f下的轨道。例如，如果f(x) = 2x+3，我们从数字1开始，那么轨道就是1、5、13、29、61，以此类推。通过允许f是一个具有多个变量的代数映射，我们也可以用数字对、三元组和n元组来形成轨道。代数方程只是我们在代数入门阶段遇到的熟悉的二次多项式的一个更一般的版本，除了它可能有更多的变量和更高的次项。这些轨道显示出许多有趣的特性。一方面，它们的结构很少，可能是生成随机数的好引擎；另一方面，人们可能希望它们的代数和算术性质可以产生新的类似于分形和混沌理论的理论，这在许多情况下是由于考虑轨道的几何和解析性质而产生的。

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.