English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We study the collection of microsets of randomly constructed fractals, whichin this paper, are referred to as Galton-Watson fractals. This is a model thatgeneralizes Mandelbrot percolation, where Galton-Watson trees (whose offspringdistribution is not necessarily binomial) are projected to $\mathbb{R}^d$ by acoding map which arises from an iterated function system (IFS) of similaritymaps. We show that for such a random fractal $E$, whenever the underlying IFSsatisfies the open set condition, almost surely the Assouad dimension of $E$ isthe maximal Hausdorff dimension of a set in $\text{supp}\left(E\right)$, thelower dimension is the smallest Hausdorff dimension of a set in$\text{supp}\left(E\right)$, and every value in between is the Hausdorffdimension of some microset of $E$. In order to obtain the above, we firstanalyze 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 weakerproperty than homogeneity and show that for self-homothetic sets in$\mathbb{R}$ whose Hausdorff dimension is smaller than 1, it is equivalent tothe weak separation condition.

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