有限细分规则的展开性质I

Lattes映射与有限细分规则

Lattes maps and finite subdivision rules

本文讨论了将Lattes映射实现为有限细分规则的细分映射。主要结果是，除了1000个解析共轭类外，所有解析共轭类中的Lattes映射都可以实现为具有一种瓦型的有限细分规则的细分映射。给出了一个Lattes映射的例子，该映射不是有限细分规则的细分映射，并且i）具有两个瓦片类型和细分复合体的1-骨架圆或ii）具有一个瓦片类型。

Expansion properties for finite subdivision rules II

We prove that every sufficiently large iterate of a Thurston map which is not doubly covered by a torus endomorphism and which does not have a Levy cycle is isotopic to the subdivision map of a finite subdivision rule. We determine which Thurston maps doubly covered by a torus endomorphism have iterates that are isotopic to subdivision maps of finite subdivision rules. We give conditions under which no iterate of a given Thurston map is isotopic to the subdivision map of a finite subdivision rule.

由有理映射构造剖分规则

Constructing subdivision rules from rational maps

本文深化了临界有限有理映射与有限细分规则之间的联系。主要定理是：如果f是一个没有周期临界点的临界有限有理映射，则对任意充分大的整数n，f^n是一个有限细分规则的细分映射。

Levy and Thurston obstructions of finite subdivision rules

For a post-critically finite branched covering of the sphere that is a subdivision map of a finite subdivision rule, we define non-expanding spines which determine the existence of a Levy cycle in a non-exhaustive semi-decidable algorithm. Especially when a finite subdivision rule has polynomial growth of edge subdivisions, the algorithm terminates very quickly, and the existence of a Levy cycle is equivalent to the existence of a Thurston obstruction. In order to show the equivalence between Levy and Thurston obstructions, we generalize the arcs intersecting obstruction theorem by Pilgrim and…

临界预周期二次配对的细分规则构造

Subdivision rule constructions on critically preperiodic quadratic matings

配对是识别多项式对的域以便在所得商空间上获得新映射的操作。交配的动力学则取决于两个多项式和商空间的定义方式，这可能很难可视化。本研究以Hubbard树与有限剖分法则为工具，探讨具有预周期临界点的二次配对。在许多情况下，这种二次对的离散参数信息可以转化为它们交配动态的拓扑信息。在这项工作中的中心定理提供的方法，明确构建细分规则，非双曲配对模型。下面我们将介绍几个例子，并与当前的文献进行联系。

圆填充、重整化与细分规则

Circle packings, renormalizations and subdivision rules

本文利用 Teichmüller 空间上剥皮映射的迭代，为满足特定细分规则的圆填充发展了重整化理论，证明重整化算子一致收缩，从而完整解决了此类圆填充的存在性与模问题，并证明了其几何不灵活性与组合刚性。