Arithmetic Difference of Middle Cantor Sets: Self-Similarity and Measure

Abstract

Let C be the space of all middle Cantor sets, it is shown that for a dense subset L of C × C × ℝ, set C _α − λC _β forms a uniformly contracting self-similar set when (C _α, C _β, λ) ∈ L. By selecting an appropriate element (C _α C _β, λ) ∈ L, we see that C _α − λC _β does not satisfy open set condition and then we calculate its Lebesgue measure that is zero.

Keywords:

• Arithmetic difference
• iterated function system
• middle Cantor sets
• Palis conjecture