Topology and Low-Dimensional Dynamics

这个项目关注的是复杂的、平面的和真实的动态。其复杂动力学部分的总体主题是使用Thurston分层来描述三次连通性轨迹边界的组合模型，扩展Fatou-Shishikura不等式，并构建Julia集的最优拓扑/组合动力学模型。在平面动力学部分，该项目试图用基于地图扩展性的工具取代复杂动力学中的全纯工具，并利用这一想法来研究被称为“扩展多模态”的平面地图。特别是，该项目将研究“c-tent”平面地图，它结合了帐篷地图的扩展属性和复二次映射的属性。因此，它们应该在二次复映射中扮演与帐篷映射在区间中扮演的角色相同的角色。在一维动力学领域，该项目在两个方向上进行研究：(1)首席研究员希望确定区间上的所有点，使它们周围的小扰动改变地图的动力学（这是稳定性的点向版本，即使对于单峰图也是新的，并且需要原始工具）；(2)进一步发展区间图、复二次图、台球的旋转理论。本项目将使用组合方法研究动力系统及其相关集合。一个复杂的平面集合（例如，一个分形集合）的适当模型应该揭示它的结构，允许人们预测它的性质。该研究还将调查动力学中的关键现象（通常很容易观察到）与不那么明显的长期现象（如未来的周期性收敛行为）之间的联系。因此，这两个主题旨在预测集合和动力系统的性质，这些信息对于理解许多具体的物理现象是必要的。此外，还将探讨各种平面动力系统之间的联系，例如局部由收缩/膨胀和旋转组成的系统或局部扩展点之间距离的系统。首席研究员还将研究这样一些点，即它们周围的小扰动会极大地改变底层系统，以及作为比旋转动力系统更复杂方案的一部分，系统表现出旋转特性的周期点。所有这些主题都需要新的技术，可以作为论文的基础，从而为人力资源的开发奠定基础。

The project is concerned with complex, plane, and real dynamics. The overarching theme of its complex dynamical part is using Thurston laminations to describe the combinatorial model for the boundary of the cubic connectedness locus, to extend the Fatou-Shishikura inequality, and to construct the finest topological/combinatorial dynamical models for Julia sets. In the part on plane dynamics the project seeks to replace holomorphic tools from complex dynamics with tools based upon the expansiveness of maps and use this idea to study plane maps called "expanding polymodials." In particular, the project will investigate "c-tent" plane maps that combine the expansive properties of tent maps with properties of complex quadratic maps. As such, they should play the same role for quadratic complex maps that tent maps play on the interval. In the realm of one-dimensional dynamics, the project pursues research in two directions: (1) the principal investigator wants to determine all points on the interval such that small perturbations around them change the dynamics of the map (this serves as a pointwise version of stability, is new even for unimodal maps, and requires original tools); (2) the project will develop further the rotation theory for interval maps, complex quadratic maps, and billiards.This project will use combinatorial methods to study dynamical systems and their related sets. An appropriate model of an intricate planar set (e.g., a fractal set) should shed light upon its structure, allowing one to predict its properties. The research will also investigate the connection between critical phenomena in dynamics (normally easy to observe) and long-term phenomena that are not so obvious (such as convergence to periodic behavior in the future). Thus, these two topics aim at predicting properties of sets and dynamical systems, information that is necessary for understanding many concrete physical phenomena. In addition, the connection between various types of planar dynamical systems will be explored, such as systems that are locally compositions of shrinking/expansion and rotation or systems that locally expand the distance between points. The principal investigator will also study points such that small perturbations around them drastically change the underlying system, as well as periodic points at which a system exhibits rotational properties, as part of a more complicated scheme than a rotational dynamical system. All these topics require new techniques and can be used as a basis for theses, and thus for the development of human resources.