On microsets, Assouad dimension and lower dimension of random fractals, and Furstenberg's homogeneity

We study the collection of microsets of randomly constructed fractals, which in this paper, are referred to as Galton-Watson fractals. This is a model that generalizes Mandelbrot percolation, where Galton-Watson trees (whose offspring distribution is not necessarily binomial) are projected to $\mathbb{R}^d$ by a coding map which arises from an iterated function system (IFS) of similarity maps. We show that for such a random fractal $E$, whenever the underlying IFS satisfies the open set condition, almost surely the Assouad dimension of $E$ is the maximal Hausdorff dimension of a set in $\text{supp}\left(E\right)$, the lower dimension is the smallest Hausdorff dimension of a set in $\text{supp}\left(E\right)$, and every value in between is the Hausdorff dimension of some microset of $E$. In order to obtain the above, we first analyze the relation between the collection of microsets of a (deterministic) set, and certain limits of subtrees of an appropriate coding tree for that set. The results of this analysis are also applied, with the required adjustments, to gain some insights on Furstenberg's homogeneity property. We define a weaker property than homogeneity and show that for self-homothetic sets in $\mathbb{R}$ whose Hausdorff dimension is smaller than 1, it is equivalent to the weak separation condition.