Algebraic dynamics

这个项目的重点是代数动力学。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非多项式映射或有理映射。人们很自然地会问，一个轨道内的无限族元素可以满足什么样的代数关系。PI猜想，如果一个轨道内的无穷多个元素满足一个代数关系，那么这个关系在某种意义上一定在映射的某个迭代下是稳定的。这个猜想可以被认为是著名的关于群簇的Mordell-Lang猜想的一般动力学模拟。到目前为止，这个猜想已经在一些特殊情况下用丢番图和p-进分析相结合的方法得到了证明。这个项目的主要目的是从总体上证明这个猜想。这个项目主要研究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。对单个数字重复应用该映射可得到该数字在f下的轨道。例如，如果f(X)=2x+3，我们从数字1开始，则轨道是1，5，13，29，61，依此类推。人们还可以通过允许f是多个变量的代数映射来形成数字对、三元组和n元组之外的轨道。代数方程只不过是我们在初学代数中遇到的常见二次多项式的一个更一般的版本，只是它可能有更多的变量和更高的次数项。人们期望，如果一个轨道中的无穷多个数满足某种代数关系，那么这种关系与代数映射有内在的联系。PI对这些线提出了一个精确的猜想，他希望能证明这一点。证明的策略涉及广泛的技术，一些数论、一些测量学和一些分析。

This project focuses on the subject of algebraic dynamics. Orbits ofpoints under iterates of maps are among the most important objects indynamics. In the study of algebraic dynamics, the map is typically apolynomial or rational map in one or more variables. It is natural toask what kinds of algebraic relations infinite families of elementswithin an orbit can satisfy. The PI conjectures that if infinitelymany elements within an orbit satisfy an algebraic relation, then thatrelation must in some sense be stable under some iterate of the map.This conjecture may be thought of as a general dynamical analog of thewell-known Mordell-Lang conjecture for group varieties. Thus far,this conjecture has been proved in a few special cases using acombination of diophantine and p-adic analytic methods. The main goalof this project is to prove the conjecture in general. This project focuses on the interaction between algebraic maps andalgebraic equations. An algebraic map is a function such as f(x) =2x+3. Applying the map repeatedly to a single number gives what iscalled the orbit of that number under f. For example, if f(x) = 2x+3and 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, andn-tuples of numbers by allowing f to be an algebraic map in more thanone variable. An algebraic equation is simply a more general versionof the familiar quadratic polynomials one encounters in beginningalgebra, except that it may have more variables and have higher degreeterms. One expects that if infinitely many numbers in an orbitsatisfy some algebraic relation, then that relation has an innateconnection with the algebraic map. The PI states a precise conjecturealong these lines, which he hopes to prove. The strategy for theproof involves a wide range of techniques, some number theoretic, somegeometric, and some analytic.