Random subsets of Cantor sets generated by trees of coin flips

Abstract

We introduce a natural way to construct a random subset of a homogeneous Cantor set $C$ in $[0,1]$ via random labelings of an infinite $M$-ary tree, where $M\ge 2$. The Cantor set $C$ is the attractor of an equicontractive iterated function system $\{f_1,\dots ,f_N\}$ that satisfies the open set condition with $(0,1)$ as the open set. For a fixed probability vector $(p_1,\dots ,p_N)$, each edge in the infinite $M$-ary tree is independently labeled $i$ with probability $p_i$, for all $i=1,2,\dots ,N$. Thus, each infinite path in the tree receives a random label sequence of numbers from $\{1,2,\dots ,N\}$. We define $F$ to be the (random) set of those points $x\in C$ which have a coding that is equal to the label sequence of some infinite path starting at the root of the tree. The set $F$ may be viewed as a statistically self-similar set with extreme overlaps, and as such, its Hausdorff and box-counting dimensions coincide. We prove non-trivial upper and lower bounds for this dimension, and obtain the exact dimension in a few special cases. For instance, when $M=N$ and $p_i=1/N$ for each $i$, we show that $F$ is almost surely of full Hausdorff dimension in $C$ but of zero Hausdorff measure in its dimension. For the case of two maps and a binary tree, we also consider deterministic labelings of the tree where, for a fixed integer $m\ge 2$, every $m$th edge is labeled $1$, and compute the exact Hausdorff dimension of the resulting subset of $C$.