Interactions between number theory and complex dynamical systems

数论与复杂动力系统之间的相互作用

• 批准号: RGPIN-2017-05656
• 负责人: Ingram, Patrick
• 金额: $ 1.46万
• 依托单位: York University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=759003
• 关键词: Interactions; between; number; theory; complex

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.

动力系统在数学应用中无处不在，而离散时间系统长期以来一直使用复分析的工具进行研究。算术动力学是一个相对较新的领域，寻求把算术几何的adelic工具承担以前只考虑从复杂的全纯的角度来看问题，这是我们提出的研究的重点。我们已经成功地利用算术几何提供了新的见解，例如，后临界有限动力系统，在目前提出的计划，我们将扩大我们的研究的算术临界轨道，有关各种算术不变量的临界轨道代数几何结构的模空间。这方面的第一个主要案例是我们最近对西尔弗曼猜想的证明，即一元有理函数模空间上的临界高度（任何程度的）是相称的任何充足的韦尔高度在该空间，连接一个容易计算，并动态自然措施的复杂性（临界高度）与从模量空间角度看更自然的几何方便测量，并允许访问算术几何的传统工具。虽然西尔弗曼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。人们应该想在更高的维度上证明这个猜想的类似物（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，在单变量情况下还有更多的工作要做。特别是，西尔弗曼猜想（或可能只是其主要推论）的动机是一个刚性的结果后临界有限理性函数，由于瑟斯顿。关于稳定族的麦克马伦定理允许我们展示瑟斯顿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），而这类更广泛的结果表明了一种推广西尔弗曼猜想的自然方式，不仅考虑了临界高度，还考虑了其他相关的临界后复杂性度量。与此同时，我们最近对西尔弗曼猜想的证明可以应用于代数簇的函数域上，以改进麦克马伦的结果（尽管不是独立的证明，因为麦克马伦定理是一个输入）。更一般的算术结果，适用于函数领域，将有应用家庭的地图在复杂的全纯setting.Finally，我们将继续我们的发展算术动力学的对应关系（迭代关系，而不是功能），和应用算术动力学的研究Drinfeld模块。这两个主题与多个地方的主要课程相联系，并为各级学生提供更大范围的入学点。