Conformal Dynamics

摘要：该项目专注于两个问题，一个是实共形动力学问题，另一个是复杂问题。实数部分是关于区间的多峰映射，即具有有限多个转向点的区间的光滑映射；只有一个转向点的映射是单峰变换。在这里，提案的目标可以非常简单地概括：从过去五年如此成功的单峰理论到多模式案例，可以带来什么？复杂的问题与Siegel Disks有关。Siegel圆盘是复杂动力系统的子域，通常是有理映射，其中动力学可以通过全纯坐标变化而线性旋转。围绕着Siegel圆盘的边界有很多谜团。最早的见解是在八十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，提出了一些不属于共形动力学的研究领域，即二维系统中的诱导双曲性和被视为高维环境下的应用问题的组合优化。与前两个问题中设定的相当保守的目标不同，这应该被称为探索性研究。该提议主要涉及一维动力系统。这类系统通常来自描述可观测参数的时间变化的模型。单一动物种群的年复一年的变化导致了这种类型的系统，前提是假设下一年的种群由当年的数量决定，而不是其他因素。一些将被研究的例子，特别是Logistic家族，是从包括物理学、经济学和生态学在内的不同领域感兴趣的模型发展而来的。当转换回这些模型时，该提案所涉及的问题与此类系统的长期行为和稳定性有关。例如：人口是否会趋于长期均衡？该建议还涉及到组合优化问题。许多专家认为，如果坚持准确的解决方案，像“旅行推销员”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，目的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.