The dimension of random subsets of self-similar sets generated by branching random walk

Abstract

Given a self-similar set $\Lambda$ that is the attractor of an iterated function system (IFS) $\{f_1,\dots ,f_N\}$, consider the following method for constructing a random subset of $\Lambda$: Let $\mathbf{p}=(p_1,\dots ,p_N)$ be a probability vector, and label all edges of a full $M$-ary tree independently at random with a number from $\{1,2,\dots ,N\}$ according to $\mathbf{p}$, where $M\ge 2$ is an arbitrary integer. Then each infinite path in the tree starting from the root receives a random label sequence which is the coding of a point in $\Lambda$. We let $F\subset \Lambda$ denote the set of all points obtained in this way. This construction was introduced by Allaart and Jones [J. Fractal Geom. 12 (2025), 67--92], who considered the case of a homogeneous IFS on $\mathbb{R}$ satisfying the Open Set Condition (OSC) and proved non-trivial upper and lower bounds for the Hausdorff dimension of $F$. We demonstrate that under the OSC, the Hausdorff (and box-counting) dimension of $F$ is equal to the upper bound of Allaart and Jones, and extend the result to higher dimensions as well as to non-homogeneous self-similar sets.

30 pages, 1 figure. A mistake in the proof of Theorem 1.2 was corrected and more details were added about the phase transitions in Figure 1