Combinatorics and Dynamics of Iterated Rational Maps

迭代有理图的组合学和动力学

• 批准号: 124336066
• 负责人: Professor Dr. Dierk Schleicher
• 金额: --
• 依托单位: Institut de Mathématiques de Marseille
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2009
• 资助国家: 德国
• 起止时间: 2008-12-31 至 2014-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/124336066?language=en
• 关键词: Combinatorics; Dynamics; Iterated; Rational; Maps

The goal of this research project is a combinatorial classification of all postcritically finite rational maps that are Newton maps of polynomials. This is a large class of all rational maps of given degree, and this will provide the first classification of a family of rational maps beyond polynomials or special one-parameter families. Along the way, we plan to answer a question of Steven Smale who asked for a classification of all those Newton maps that have additional attracting cycles (that is, attracting cycles that are not the roots of the given polynomials). This classification will be done in terms of graphs that we call "Newton graphs" and that naturally occur in the dynamical plane of the Newton maps. We also intend to investigate which postcritically finite Newton maps are "matings" of two polynomials: the latter is a known method to describe the dynamics of certain rational maps in terms of two polynomials of the same degree. It is known that all cubic Newton maps can be understood in this way (together with a related method called "capture"), but in higher degrees very little is currently known. Finally, we plan to extend our classification to all those rational Newton maps that arise as Newton maps of transcendental functions: these differ from polynomial Newton maps in the way that they have a parabolic, rather than repelling, fixed point at infinity.

这个研究计画的目标是对所有的后临界有限有理映射进行组合分类，这些有理映射是多项式的牛顿映射。这是给定次数的所有有理映射的一大类，并且这将提供除多项式或特殊单参数族之外的有理映射族的第一个分类。沿着，我们计划回答史蒂芬·斯梅尔的一个问题，他要求对所有具有额外吸引圈（即吸引圈不是给定多项式的根）的牛顿映射进行分类。这种分类将根据我们称之为“牛顿图”的图来完成，这些图自然地出现在牛顿映射的动力学平面中。我们还打算调查后临界有限牛顿映射是“交配”的两个多项式：后者是一个已知的方法来描述动力学的某些合理的地图在两个多项式的相同程度。众所周知，所有的三次牛顿映射都可以用这种方式来理解（连同一种称为“捕获”的相关方法），但在更高的程度上，目前所知甚少。最后，我们计划将我们的分类扩展到所有那些作为超越函数的牛顿映射出现的有理牛顿映射：这些映射与多项式牛顿映射的不同之处在于它们在无穷远处有一个抛物线而不是排斥的不动点。