具有正不动点比例和正豪斯多夫维数的一类水平传递群
A family of level-transitive groups with positive fixed-point proportion and positive Hausdorff dimension
AI总结：
本文提出计算迭代 wreath 积不动点比例的方法，构造了一类满足特定条件的水平传递群，其不动点比例可显式计算，还计算了对应多项式迭代伽罗瓦群的不动点比例。
摘要：
本文提供了一种计算作用于d-正则树的任意迭代 wreath 积的不动点比例的方法。此外，该方法适用于作用于d-正则树的迭代 wreath 积的推广形式，这些推广形式并非群。作为该推广的一个应用，本文构造了一类作用于d-正则树的深度为2的有限型群，其中d≥3且d≢2 mod4。这些群是自相似的、水平传递的，具有正豪斯多夫维数，且呈现正不动点比例。与文献中已知的其他具有正不动点比例的群不同，该新家族的不动点比例可显式计算。此外，多项式x^d +1（d≥2）的迭代伽罗瓦群出现在该家族中，因此计算了其不动点比例。
英文摘要：
This article provides a method to calculate the fixed-point proportion of any iterated wreath product acting on a $d$-regular tree. Moreover, the method applies to a generalization of iterated wreath products acting on a $d$-regular tree, which are not groups. As an application of this generalization, a family of groups of finite type of depth $2$ acting on a $d$-regular tree with $d \geq 3$ and $d \neq 2 \pmod{4}$ is constructed. These groups are self-similar, level-transitive, have positive Hausdorff dimension, and exhibit a positive fixed-point proportion. Unlike other groups with a positive fixed-point proportion known in the literature, the fixed-point proportion of this new family can be calculated explicitly. Furthermore, the iterated Galois group of the polynomial $x^d + 1$ with $d \geq 2$ appears in this family, so its fixed-point proportion is calculated.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
