Rubber Bands to Rational Maps

多年来，复数多项式和有理映射的动力学一直是数学家们的兴趣和灵感。他们的Julia集合提供了分形的原型例子，并展示了从非常简单的规则开始的迷人的动态行为。在这个项目中，PI将致力于更好地理解分形的几何，特别是那些由“有理映射”生成的分形，推广Mandelbrot集合中使用的多项式。PI已经开发了一种简单的地图描述，使用图形，被认为是橡皮筋网络。这比以前的方法更容易获得，降低了本科生和研究生的参与障碍，让他们亲身体验和了解美丽图片背后的结构。特别地，这个项目将允许对最简单的有理图进行分类，以及一个新的测试，当一个松散的（拓扑的）有理图的描述可以被收紧到产生分形的刚性图像中时。用更专业的术语来说，该项目将研究从球体到自身的某些映射（拓扑分支的自覆盖），这些映射与理性映射相关。要确定理性映射的行为，要查看的关键点是关键点，即映射分支的位置。映射可以通过给球体上的后临界集的补和它的逆像一个适当的脊来描述。一个关键的问题是，当一个分支的自覆盖可以“刚性”地变成一个理性的映射时，要进行表征。威廉·瑟斯顿（William Thurston）在1982年给出了一个答案，就某些障碍物而言，环空的集合不可能存在于实际的理性地图中。这个答案陈述起来很复杂，并且很难应用，但是它仍然提供了构建理性映射的少数工具之一。在这个项目中，PI将研究一个补充标准，它松散地说脊柱（边缘有弹性重量）在重复的向后迭代下变得更松。新的标准承诺既简单又强大，PI将与其他研究人员、研究生和本科生一起探索它。例如，他们将利用新的特征来构建一个新的理性地图普查。他们还将研究主要定理的扩展，并研究它揭示的新结构。

The dynamics of complex polynomials and rational maps has been intriguing and inspiring mathematicians for many years. Their Julia sets provide prototypical examples of fractals and exhibit fascinating dynamical behavior starting from very simple rules. In this project, the PI will work to better understand the geometry of fractals, particularly those generated by "rational maps", generalizing the polynomials used in the Mandelbrot set. The PI has developed a simple description of such maps using graphs, thought of as networks of rubber bands. This promises to be more accessible than previous approaches, lowering the barrier to participation by undergraduates and beginning graduate students, and letting them experience and understand first-hand the structure behind the beautiful pictures. In particular, the project will allow for a classification of the simplest rational maps, and a new test for when a loose (topological) description of a rational map can be tightened up into the rigid picture that generates the fractal.In more technical terms, the project will study certain maps from the sphere to itself (topological branched self-covers) that are related to rational maps. To determine the behavior of the rational map, the key points to look at are the critical points, where the map branches. A map can be described by giving an appropriate spine for the complement of the post-critical set on the sphere, along with its inverse image. One key question is to characterize when a branched self-cover can be made "rigid" into a rational map. William Thurston gave one answer in 1982, in terms of certain obstructions, collections of annuli that could not exist inside actual rational maps. This answer is complicated to state and difficult to apply, but it nevertheless provides one of the few tools for constructing rational maps. In this project, the PI will study a complementary criterion, which loosely says that the spine (with elastic weights on the edges) get looser under repeated backwards iteration. The new criterion promises to be both simple and powerful, and the PI will work with other researchers, graduate students, and undergraduate students to explore it. They will use the new characterization to, for instance, construct a new census of rational maps. They will also investigate extensions of the main theorem and study the new structure it reveals.