Bifurcations in Complex Algebraic Dynamics

从理论、计算或实践的角度来看，动力系统的稳定性可以说是其最重要的特征。对于随时间演化的系统，我们的目标是确定哪些扰动将保持系统的长期行为，哪些扰动将导致完全不同的结果。本计画主要研究复代数动力系统的稳定性与分叉。这样的系统由一个或多个变量的多项式公式定义。定义方程的代数性质将动力学研究与代数几何的丰富理论联系起来。此外，在所有定义多项式都具有整数系数的例子中，相关的动力学稳定性问题与数论和基本方程的丢番图几何有着惊人的联系。该项目将扩展复杂的分析实例的动态稳定性理论，以自然出现在算术几何和复杂动力学的新设置。该项目还为研究生和博士后提供研究和培训机会。该项目开发了射影空间上的解析映射族的稳定性理论，无论是在复解析环境中还是在非阿基米德值域和p-adic分析环境中。在PI和其他研究人员的早期工作中，最近发现可以使用复杂动力学来分析高度函数和算术交叉理论的某些问题。在一系列最近的突破算术几何，特别是关于统一的界限，一些合理的家庭点的代数簇，稳定性理论发挥了至关重要的-如果有些隐藏-的作用。该项目旨在阐明稳定性理论的作用，并进一步推动该理论。许多提出的问题和应用的理论有关的发生“不太可能的交叉点”在家庭的阿贝尔品种或更一般的家庭极化动力系统。该项目的具体目标包括：（1）表征某些分叉电流和测度的正性性质;（2）为代数动力系统提供不变子簇几何上的界限;（3）在p进解析映射族的背景下制定分叉理论。根据该奖项进行的研究活动预计将影响数学的多个领域，包括数论，几何和动力学。该奖项反映了NSF的法定使命，并已被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

The stability of a dynamical system is arguably its most important feature, from a theoretical, computational, or practical point of view. For systems that evolve with time, one aims to determine which perturbations will preserve the system’s long-term behavior and which perturbations will lead to radically different outcomes. This project concerns stability and bifurcations in the setting of complex algebraic dynamical systems. Such systems are defined by polynomial formulas in one or several variables. The algebraic nature of the defining equations connect the dynamical study with the rich theory of algebraic geometry. Moreover, in the case of examples where all of the defining polynomials have, for example, integer coefficients, the relevant dynamical stability questions have surprising connections to number theory and to the Diophantine geometry of the underlying equations. The project will extend the theory of dynamical stability for complex analytic examples to new settings that arise naturally in arithmetic geometry and complex dynamics. The project also provides research and training opportunities for graduate students and postdocs.This project develops the theory of stability for analytic families of maps on projective spaces, in both a complex analytic setting and in the setting of non-archimedean-valued fields and p-adic analysis. It was recently discovered, in earlier work of the PI and of other researchers, that certain questions about height functions and arithmetic intersection theory can be analyzed using complex dynamics. In a series of recent breakthroughs in arithmetic geometry, especially concerning uniform bounds for numbers of rational points on families of algebraic varieties, stability theory played a crucial--if somewhat hidden--role. This project aims to shed new light on the role of stability theory and to push the theory further. Many of the proposed problems and applications of the theory are related to the occurrence of `unlikely intersections’ in families of abelian varieties or in more general families of polarized dynamical systems. Specific goals of this project include (1) to characterize positivity properties of certain bifurcation currents and measures; (2) to provide bounds on the geometry of invariant subvarieties for algebraic dynamical systems; and (3) to formulate a theory of bifurcations in the setting of p-adic analytic families of maps. The research activity conducted under this award is expected to impact multiple areas of mathematics, including number theory, geometry, and dynamics.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.