Multiple codings of self-similar sets with overlaps

In this paper we consider a general class E of self-similar sets with complete overlaps. Given a self-similar iterated function system Φ=(E,{f_i}_i=1^m)∈E on the real line, for each point x∈E we can find a sequence (i_k)=i₁i₂…∈{1,…,m}^N, called a coding of x, such that x=limn→∞⁡f_i1∘f_i2∘⋯∘f_in(0). For k=1,2,…,ℵ₀ or 2^ℵ₀ we investigate the subset U_k(Φ) which consists of all x∈E having precisely k different codings. Among several equivalent characterizations we show that U₁(Φ) is closed if and only if U_ℵ0(Φ) is an empty set. Furthermore, we give explicit formulae for the Hausdorff dimension of U_k(Φ), and show that the corresponding Hausdorff measure of U_k(Φ) is always infinite for any k≥2. Finally, we explicitly calculate the local dimension of the self-similar measure at each point in U_k(Φ) and U_ℵ0(Φ).