A family of level-transitive groups with positive fixed-point proportion and positive Hausdorff dimension

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 $d\ge 3$ and $d\mathrm{\ne }2(\mathrm{m}\mathrm{o}\mathrm{d}4)$ 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 $d\ge 2$ appears in this family, so its fixed-point proportion is calculated.