Some problems in arithmetic dynamics and related areas

动力系统在现代数学中无处不在。一个离散动力系统是一个集合S和一个从S到它自身的映射f，而动力学是对迭代族{f，f2=f o f，f3=f o f，.}的研究。例如，当S是真实的数的集合，f是将数发送到其平方的映射时，迭代族变为{f（x）=x2，f2（x）=x4，f3（x）=x8，.}。我建议研究的情况下，S是一个对象定义的多项式方程（即代数簇）和映射f可以描述的多项式（即代数态射）。这项建议有两个部分。 第一部分涉及不太可能相交的原则，即如果两个物体的相交大于预期，则应该有潜在的几何原因。作为最简单的例子，我们有，如果两条线的交点包含一个以上的点，那么这些线重合。这个有点天真的原则是一个重要的驱动力背后的重大发展丢番图几何和算术动力学在过去10年。在许多结果不太可能的交叉动力学，我得到的，最近的一个是一个新的有界高度现象的动力学最近的工作Amoroso，Masser和Zannier。我建议改进我早期工作中的技术，并发现新的策略来证明更多动力系统的有界高度结果。此外，我相信在上述工作中使用的技术将有更多的应用到算术动力学的其他问题，我也计划研究这样的应用。 第二部分是算术动力学的结果和思想在相关领域的应用。这说明算术动力学不仅本身很有趣，而且在其他领域也有非常有趣的问题。有两种“类型”的应用程序，我希望研究：第一种类型涉及“算术部分”，而第二种类型涉及“动力学部分”的算术动力学。更具体地说，对于第一种类型，我建议证明代数独立的某些马勒函数发挥了重要作用，超越数论自20世纪70年代以来。我提出的研究可能有助于解决马勒理论中关于这类函数的“非消失步骤”这一众所周知的难题。对于第二种类型，我的目标是发现算术动力学和其他动力系统理论之间的新联系和类比。算术动力学是一个年轻的领域，在某种意义上，它直接起源于丢番图几何和复动力学。另一方面，有几个完善的理论，如拓扑动力学，遍历理论，符号动力学（与应用信息论）等动力系统，虽然这第二个方面是不太明确的时刻，它有可能开辟高度有趣的研究方向算术动力学。

Dynamical systems are ubiquitous in modern mathematics. A discrete dynamical system is a set S together with a map f from S to itself, and dynamics is the study of the family of iterates {f, f2=f o f, f3=f o f o f,...}. For example, when S is the set of real numbers and f is the map that sends a number to its square, the family of iterates becomes {f(x)=x2, f2(x)=x4, f3(x)=x8,...}. I propose to study the situation when S is an object defined by polynomial equations (i.e. an algebraic variety) and the map f can be described by polynomials (i.e. algebraic morphism). This proposal has two parts. The first part involves the principle of unlikely intersections suggesting that if the intersection of two objects is larger than expected, there should be an underlying geometric reason. As the simplest example, we have that if the intersection of two lines contains more than one point then the lines coincide. This somewhat naive principle is an important driving force behind significant developments in diophantine geometry and arithmetic dynamics in the last 10 years. Among many results on unlikely intersections in dynamics that I obtained, the most recent one is a new bounded height phenomenon in dynamics motivated by recent work of Amoroso, Masser, and Zannier. I propose to improve the techniques in my earlier work and discover new strategies to prove bounded height results for many more dynamical systems. In addition, I believe that the techniques used in the above work will have more applications to other problems in arithmetic dynamics and I also plan to study such applications. The second part is about applications of results and ideas in arithmetic dynamics to related areas. This illustrates that arithmetic dynamics is not only interesting on its own but also has applications to highly fascinating problems in other areas. There are two "types" of applications that I wish to study: the first type involves the "arithmetic part" while the second type involves the "dynamics part" of arithmetic dynamics. More specifically, for the first type, I propose to prove algebraic independence of certain Mahler functions which played a significant role in transcendental number theory since the 1970s. My proposed research might help settle the notoriously difficult "non-vanishing step" in Mahler's theory for such functions. For the second type, I aim to discover new connections and analogies between arithmetic dynamics and other theories of dynamical systems. Arithmetic dynamics is a young area, and in a certain sense, descended directly from diophantine geometry and complex dynamics. On the other hand, there are several well established theories concerning dynamical systems such as topological dynamics, ergodic theory, symbolic dynamics (with applications to information theory), etc. Although this second aspect is less definite at the moment, it has the potential to open up highly interesting research directions in arithmetic dynamics.