Random subsets of Cantor sets generated by trees of coin flips

TL;DR

This paper introduces a probabilistic method to generate random subsets of Cantor sets using infinite trees with random labelings, analyzing their fractal dimensions and revealing conditions for full or zero measure.

Contribution

It develops bounds and exact formulas for the Hausdorff and box-counting dimensions of these random Cantor subsets, including special cases with uniform and deterministic labelings.

Findings

Random subsets can have full Hausdorff dimension in the original Cantor set.

Exact dimensions are computed for specific label distributions and tree structures.

The set's Hausdorff and box-counting dimensions coincide despite overlaps.

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 $M\geq 2$. 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 $i=1,2,\dots,N$. 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 $m\geq 2$, every $m$th edge is labeled $1$, and compute the exact Hausdorff dimension of the resulting subset of $C$.