Examples for complex dynamics in several variables

这个研究项目的重点是两个或更多复杂变量的理性映射的动力学。这样的地图已经被研究了大约20年，具体的例子在发展一个更普遍的理论中起着至关重要的作用。在这些例子的基础上，有一个清晰的猜想，描述了人们应该从这些地图的“最不稳定”的动力学部分中期待什么，即支持最大熵的度量。对于两个变量的有理映射，已经取得了重大进展，猜想的验证几乎完成。远离这种最大熵度量的支持的动力学也很有意义。在某些情况下，存在一个物理上相关的不变实片，它不支持此度量。在其他情况下，人们可能会对Fatou集合的拓扑感兴趣，在该拓扑上映射的动态是稳定的。对于这些类型的问题，我们需要新的猜想，PI认为通过研究适当的例子可以最好地发展这些猜想。PI提出了三个不同的项目，每个项目都提供了可以激发在几个变量中复杂动力学的更深层次理论的例子。它们是：(1)研究作为分层格上Ising模型（磁性材料模型）的重整化算子产生的有理映射，(2)研究复射影平面的全局全纯映射的Fatou分量，以及(3)研究具有不定点的后临界有限有理映射族的第一性质。所有三个被提议的项目都将在几个变量中为特定的理性映射的动态行为制定细节。这将推动当代理论的发展，也有助于发展新的猜想。此外，与Ising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的更深入的理论理解。这有可能产生更广泛的影响，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交流。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积极与之合作）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.