A Class of Self-Similar Fractals with Overlap Structure

AbstractFor the contracting similaritiesS1(x)=x/3,S2(x)=(x+λ)/3, andS3(x)=(x+2)/3, where λ∈[0,1], letFλdenote the invariant set with respect toS1,S2, andS3. In this paper, we study the Hausdorff measure, Hausdorff dimension, and the structure ofFλ. Let λ=b/a∈Q∩[0,1], (a,b)=1. Using combinatorial techniques, we show that |Fλ|>0 ifa≡b≢0 (mod3); otherwise,Fλis a Cantor-like fractal with recursive construction