Properties of a random Cantor set with overlaps

We study the topology and the Hausdorff dimension of a random Cantor set with overlaps, generated by an iterated function system with scaling ratio equal to the Golden Mean. The results extend known formulas to a case where the Open Set Condition fails. Our methodology is based on the theory of expansions in non-integer bases.

💡 Research Summary

The paper investigates a random Cantor set generated by an iterated function system (IFS) consisting of two affine contractions f₀(x)=x/φ and f₁(x)=(x+1)/φ, where φ=(1+√5)/2 is the golden mean. Unlike the classical setting, the IFS does not satisfy the Open Set Condition (OSC) because the similarity dimension log 2 / log φ exceeds the ambient dimension 1, leading to exact overlaps: different binary words can produce the same interval (e.g., I₁₁₀ = I₀₀₁). To handle this, the authors work with greedy φ‑expansions, which are the lexicographically largest binary representations that avoid the forbidden word “011”. Each interval is associated with a greedy word, and all words representing the same interval form an equivalence class