Some problems in arithmetic dynamics and related areas

动力系统在现代数学中无处不在。离散动力系统是一个集合S加上从S到它自己的映射f，而动力学是研究迭代族{f，f2=f f，f3=f f，…}。例如，当S是实数集，f是将一个数送到它的平方的映射时，迭代族变成{f(X)=x2，f2(X)=x4，f3(X)=x8，...}。我建议研究当S是由多项式方程(即代数簇)定义的对象，而映射f可以用多项式(即代数态射)来描述时的情况。这项提议包括两个部分。 第一部分涉及不太可能相交的原理，这表明如果两个物体的相交比预期的大，应该有潜在的几何原因。最简单的例子是，如果两条直线的交点包含一个以上的点，则这两条直线重合。这个有点幼稚的原理是过去10年丢番图几何和算术动力学重大发展背后的重要推动力。在我获得的许多关于动力学中不可能相交的结果中，最新的一个是动力学中一种新的有界高度现象，这一现象是由Amoroso、Masser和Zannier最近的工作推动的。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，我相信上述工作中使用的技术将会在算术动力学中的其他问题上有更多的应用，我也计划研究这些应用。 第二部分是关于算术动力学的结果和思想在相关领域的应用。这表明算术动力学不仅本身很有趣，而且还可以应用到其他领域非常有趣的问题上。我希望研究两种“类型”的应用：第一种涉及“算术部分”，第二种涉及算术动力学的“动力学部分”。更具体地说，对于第一类，我建议证明某些Mahler函数的代数无关性，这些函数自20世纪70年代以来在超越数论中发挥了重要作用。我提出的研究可能有助于解决马勒关于这类函数的理论中出了名的困难的“不消失的一步”。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学和复杂动力学。另一方面，关于动力系统有几个成熟的理论，如拓扑动力学、遍历理论、符号动力学(及其在信息论中的应用)等。虽然这第二个方面目前不太确定，但它有可能在算术动力学中开辟非常有趣的研究方向。

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.