Multiple codings of self-similar sets with overlaps

Abstract

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,{fi}i=1m)∈E on the real line, for each point x∈E we can find a sequence (ik)=i1i2…∈{1,…,m}N, called a coding of x, such that x=limn→∞⁡fi1∘fi2∘⋯∘fin(0). For k=1,2,…,ℵ0 or 2ℵ0 we investigate the subset Uk(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 Uℵ0(Φ) is an empty set. Furthermore, we give explicit formulae for the Hausdorff dimension of Uk(Φ), and show that the corresponding Hausdorff measure of Uk(Φ) is always infinite for any k≥2. Finally, we explicitly calculate the local dimension of the self-similar measure at each point in Uk(Φ) and Uℵ0(Φ).