Examples for complex dynamics in several variables

这个研究项目的重点是两个或多个复变量的有理映射的动力学。这种地图已经被研究了大约20年，具体的例子在更普遍的理论的发展中发挥了至关重要的作用。主要基于这些例子，有一个明确的猜想，描述了人们应该从这些映射的动力学的“最不稳定”部分期待什么，即最大熵测度的支持。 已经取得了重大进展和验证的猜想是几乎完成的合理的地图在两个变量。 远离这种最大熵度量的动力学也是非常有趣的。在某些情况下，有一个物理上相关的不变的真实的切片，它远离这个度量的支持。 在其他情况下，人们可能会对Fatou集的拓扑感兴趣，在Fatou集上，映射的动力学是稳定的。对于这些类型的问题，我们需要新的知识，PI认为通过研究适当的例子可以最好地开发。PI提出了三个不同的项目，每个项目都提供了可以激发多变量复杂动力学更深入理论的例子。 它们是：（1）研究作为伊辛模型重整化算子的有理映射（2）研究复射影平面上全局全纯映射的Fatou分支，以及（3）研究一族具有不定点的后临界有限有理映射的第一性质。所有三个拟议项目都将为多元有理映射的动力学行为。这将推动当代理论的发展，也有助于发展新的理论。 此外，与伊辛模型相关的项目将有助于在三个领域（真实动力学，复杂动力学和统计物理学）之间建立新的联系，同时可能导致对磁性材料更深入的理论理解。 有更广泛的影响，特别是通过促进动力系统和数学物理研究人员之间更好的沟通显着的潜力。 与这些项目相关的工作可以影响各个层次的学生，从非常有才华的高中生（PI积极与他们合作）到博士。候选人，让他们接触到跨学科的当代研究。

This research project focuses on the dynamics of rational maps in two or more complex variables. Such maps have been studied for about twenty years, with specific examples playing a crucial role in the development of a more general theory. Based largely on these examples, there is a clear conjecture describing what one should expect from the "most unstable" part of the dynamics for these maps, i.e. the support of the measure of maximal entropy. Significant progress has been made and verification of the conjecture is almost complete for rational maps in two variables. The dynamics away from the support of this measure of maximal entropy is also of significant interest. In certain cases, there is a physically relevant invariant real slice that is away from the support of this measure. In other cases, one may be interested in the topology of the Fatou set, on which the dynamics of the map is stable. For these types of questions, we need new conjectures, which the PI believes will be best developed by studying appropriate examples. The PI proposes three different projects, each providing examples that can motivate a deeper theory of complex dynamics in several variables. They are: (1) study of rational maps arising as renormalization operators for the Ising model (a model for magnetic materials) on hierarchical lattices, (2) study of Fatou components for globally holomorphic maps of the complex projective plane, and (3) investigation of the first properties for a family of postcritically finite rational maps that have indeterminate points.All three of the proposed projects will work out details for the dynamical behavior of specific rational maps in several variables. This will push the contemporary theory and also help to develop new conjectures. Moreover, the project related to the Ising model will help to build new connections between three fields (real-dynamics, complex dynamics, and statistical physics), while potentially leading to a deeper theoretical understanding of magnetic materials. There is significant potential for broader impact, particularly by fostering better communication between researchers in dynamical systems and mathematical physics. Work related to these projects can influence students of all levels, from very talented high school students (with whom the PI actively works) through Ph.D. candidates, by exposing them to contemporary research that crosses between disciplines.