Conformal Dynamics

摘要：本课题主要研究两个问题，一个是真实保角动力学问题，另一个是复杂保角动力学问题。实部是关于区间的多模态映射，即具有有限多个拐点的区间的光滑映射；只有一个转折点的映射是单峰变换。在这里，建议的目标可以非常简单地总结：从过去五年如此成功的单模态理论，到多模态情况，可以继承什么？这个复杂的问题是关于西格尔磁盘的。西格尔盘是一个复杂动力系统的子域，通常是一个有理映射，其中的动力学可以通过全纯的坐标变化来实现线性旋转。围绕着西格尔盘的边界有很多谜团。最初的见解是在80年代中期赫尔曼先生的工作中获得的。该项目建议改进他工作中使用的一些分析工具，以获得更多信息。最后，提出了一些不属于保形动力学的研究领域，即二维系统的诱导双曲性和组合优化作为一个非常高维设置的应用问题。与前两个问题中设定的相当保守的目标不同，这些应该被称为探索性研究。该建议主要涉及一维动力系统。这样的系统通常来自于描述可观测参数随时间变化的模型。如果假设下一年的动物数量是由当年的数量而不是其他因素决定的，那么单一动物种群的年复一年变化就会导致这种类型的系统。一些将被研究的例子，特别是逻辑家族，来自不同领域的兴趣模型，包括物理学，经济学和生态学。当转换回这些模型时，提案所解决的问题与这些系统的长期行为和稳定性有关。例如：人口会趋向于长期均衡吗？本文还讨论了组合优化问题。许多专家认为，如果一个人坚持要求精确的答案，像“旅行推销员”问题这样的实际问题是难以解决的。该提案将研究一种基于热力学思想的新方法，目标是获得高性能的近似解。

Summary: The project concentrates on two problems, one in real conformal dynamics and one in complex. The real part is about poly-modal mappings of the interval, i.e., smooth maps of the interval with finitely many turning points; a map with only one turning point is a unimodal transformation. Here the goal of the proposal can be very simply summarized: what can be carried ovep from the unimodal theory, so successful in the last five years, to the poly-modal case? The complex problem is about Siegel disks. Siegel disks are sub-domains of a complex dynamical system, usually a rational map, where the dynamics can be brought to a linear rotation by a holomorphic change of coordinates. A lot of mystery has surrounded the boundaries of Siegel disks. First insights were gained only in the mid-eighties with the work of M. Herman. The project proposes to refine some of the analytic tools used in his work to get more information. Finally, some research is proposed in areas that do not belong to conformal dynamics, namely induced hyperbolicity in two-dimensional systems and combinatorial optimization viewed as an applied problem in a very high-dimensional setting. Unlike the rather conservative goals set in the first two problems, these should be termed exploratory research. The proposal is concerned mostly with one-dimensional dynamical systems. Such systems generally come from models which describe time changes of an observable parameter. Year-to-year changes in a single animal population lead to this type of system, provided one assumes that the population in the next year is determined by the current-year count and no other factors. Some examples that will be studied, notably the logistic family, grew from models of interest in diverse areas including physics, ecmnomics and ecology. The questions that the proposal addresses, when translated back to these models, are related to the long-term behavior and stability of such systems. For example: will the population tend to a long-t erm equilibrium? The proposal is also concerned with the problem of combinatorial optimization. Practical problems such as the "traveling salesman" problem are believed by many experts to be intractable if one insists on the exact solution. The proposal will examine a new approach based on the ideas of thermodynamics with the goal of getting high-performance approximate solutions.