Combinatorics and Dynamics of Iterated Rational Maps

这个研究项目的目标是对所有后临界有限有理映射进行组合分类，这些映射是多项式的牛顿映射。这是所有给定次数的有理映射族的一大类，这将提供超出多项式或特殊的单参数族的有理映射族的第一个分类。在这个过程中，我们计划回答Steven Smear的一个问题，他要求对所有具有额外吸引圈(即吸引不是给定多项式的根的圈)的牛顿映射进行分类。这种分类将根据我们称为“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。我们还打算研究哪些后临界有限牛顿映射是两个多项式的“匹配”：后者是用两个相同次数的多项式来描述某些有理映射的动力学的已知方法。众所周知，所有的三次牛顿映射都可以这样理解(连同一种相关的称为“捕捉”的方法)，但在更高的程度上，目前对此知之甚少。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而产生的有理牛顿映射：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具有抛物线而不是排斥不动点。

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.