Interactions between number theory and complex dynamical systems

动力系统在数学应用中无处不在，离散时间系统长期以来一直使用复分析工具进行研究。算术动力学是一个相对较新的领域，寻求将算术几何的强大工具应用于以前仅从复全纯角度考虑的问题，这是我们提出的研究重点。我们已经在使用算术几何方面取得了一些成功，例如，在后临界有限动力系统方面提供了新的见解。******在目前提出的方案中，我们将扩展我们对关键轨道算法的研究，将关键轨道的各种算法不变量与模空间上的代数几何结构联系起来。这方面的第一个主要案例是我们最近对Silverman猜想的证明，即在一个变量（任何程度）的有理函数的模空间上的临界高度与该空间上的任何充足的韦尔高度相称，将一个易于计算的、动态自然的复杂性度量（临界高度）与一个从模空间的角度来看更自然的几何上方便的度量（临界高度）联系起来，并且允许访问传统的算术几何工具。虽然西尔弗曼猜想的证明是算术动力学的重要一步，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的情况，但是一般的猜想要远得多）。与此同时，在单变量情况下还有更多的事情要做。特别地，西尔弗曼猜想（或者可能只是它的主要推论）是由后批判有限有理函数的刚性结果激发的，由于瑟斯顿。关于稳定家族的McMullen定理允许人们展示Thurston的刚性定理作为一类刚性结果中的一个例子（公平地说，是一个中心例子），这类更广泛的结果提出了一种推广Silverman猜想的自然方法，不仅考虑临界高度，而且考虑其他相关的后临界复杂性度量。同时，我们最近对Silverman猜想的证明可以应用于代数变量的函数域，从而给出McMullen结果的一个改进（尽管不是一个独立的证明，因为McMullen定理是一个输入）。更一般的算术结果，应用于函数域，将有应用于族映射在复全纯设置。******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模块研究中的应用。这两个主题在多个地方与主课程相关联，为各个层次的学生提供了更大范围的入门点。

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.