Random subsets of Cantor sets generated by trees of coin flips

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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 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\geq 2 , every m th edge is labeled 1 , and compute the exact Hausdorff dimension of the resulting subset of C .