Interactions between number theory and complex dynamical systems

动力系统在数学应用中是无处不在的，而离散时间系统长期以来一直是用复分析的工具来研究的。算术动力学是一个相对较新的领域，寻求将算术几何的高级工具应用于以前仅从复全纯角度考虑的问题，这也是我们提出的研究重点。我们已经成功地利用算术几何提供了一些新的见解，例如，后临界有限动力系统。*在当前提出的程序中，我们将扩展我们对临界轨道算术的研究，将临界轨道的各种算术不变量与模空间上的代数几何结构联系起来。第一个主要情况是我们最近对Silverman猜想的证明，即一元有理函数的模空间上的临界高度(任意次)与该空间上的任何充足的Weil高度相称，将一个容易计算的、动态的自然复杂性度量(临界高度)与一个几何上方便的度量联系起来，从模空间的角度来看，这是更自然的，并允许访问传统的算术几何工具。虽然Silverman猜想的证明是算术动力学向前迈出的重要一步，但它也只是朝着某个方向迈出的第一步。人们应该想要在更高的维度上证明这个猜想的类似(我们已经证明了特殊情况，但一般的猜想要远得多)。与此同时，在单变量的情况下，还有更多的事情要做。特别是，Silverman的猜想(或者可能只是它的主要推论)是由后临界有限有理函数的刚性结果推动的，这是由于瑟斯顿。麦克马伦关于稳定族的定理允许人们展示瑟斯顿的刚性定理作为一类刚性结果中的一个例子(公平地说，这是一个中心例子)，并且这类更广泛的结果建议了一种自然的方式来推广西尔弗曼的猜想，不仅考虑到临界高度，而且考虑到后临界复杂性的其他相关度量。同时，我们最近对Silverman猜想的证明可以应用到代数族的函数域上，从而改进了McMullen的结果(尽管不是一个独立的证明，因为McMullen定理是一个输入)。应用于函数域的更一般的算术结果将应用于复全纯背景下的映射族。*最后，我们将继续发展对应的算术动力学(迭代关系而不是函数)，以及算术动力学在Drinfeld模研究中的应用。这两个主题与多个地方的主课程相结合，为所有级别的学生提供了更广泛的切入点。

Dynamical systems are ubiquitous in applications of mathematics, and discrete-time systems have long been studied using the tools of complex analysis. Arithmetic dynamics is a relatively new field, seeking to bring the adelic tools of arithmetic geometry to bear on problems previously considered only from the complex holomorphic perspective, and this is the focus of our proposed research. Already we have had some success in using arithmetic geometry to provide new insight on, for example, post-critically finite dynamical systems.******In the current proposed program, we will expand our study of the arithmetic of critical orbits, relating various arithmetic invariants of critical orbits to algebro-geometric structures on the moduli space. The first major case of this is our recent proof of Silverman's Conjecture that the critical height on the moduli space of rational functions in one variable (of any degree) is commensurate to any ample Weil height on that space, connecting an easily-computed, and dynamically natural measure of complexity (the critical height) with a geometrically convenient measure which is more natural from the moduli space perspective, and allows access to the traditional tools of arithmetic geometry. Although the proof of Silverman's Conjecture is significant step forward for arithmetic dynamics, it is also only the first step in a certain direction. One should like to prove the analogue of this conjecture in higher dimensions (we have proven special cases, but the general conjecture is much further off). At the same time, there is more to do in the single-variable case. In particular, Silverman's Conjecture (or perhaps only its main corollary) is motivated by a rigidity result for post-critically finite rational functions, due to Thurston. McMullen's Theorem on stable families allows one to exhibit Thurston's rigidity theorem as one example among a class of rigidity results (in fairness, a central example), and this broader class of results suggests a natural way to generalize Silverman's Conjecture, considering not just the critical height but other related measures of post-critical complexity. At the same time, our recent proof of Silverman's Conjecture can be applied over function fields of algebraic varieties to give a refinement of McMullen's result (although not an independent proof, since McMullen's Theorem is an input). More general arithmetic results, applied over function fields, will have applications for families of maps in the complex holomorphic setting.******Finally, we will continue our development of the arithmetic dynamics of correspondences (iterating relations rather than functions), and of applications of arithmetic dynamics to the study of Drinfeld modules. These two topics tie in to the main program in multiple places, and offer a far greater range of entry points for students at all levels.