Self-similar Subsets Of The Fifth Cantor Set

In 1998, Mattila had raised such a question:what are the self-similar subsets of the middle-third Cantor set? De-Jun Feng, Hui Rao, and Yang Wang[6] have solved the question. In this paper, we will study the structures of the self-similar subsets of the fifth Cantor set.Let K5 denote the fifth Cantor set, which is the attractor of the iterated function system(IFS) T={t0,t1,t2}, where to(x)= x/5, t1(x)=x+2/5, t2(x)=x+4/5. Suppose F is the non-trivial self-similar set of IFS &, and F C K5, our goal is to study the structure of F, more accurately, to study the structure of (?).First, we introduce that the contraction ratios of the maps included in (?) can be written as ±5-m, where m ∈ N. And we prove that every self-similar subset of K5 has a normal expression, that is F can be written as α+FΦ, where α ∈ K5,Φ ∈ F, and F is a special class of IFS’.Second, we introduce a method to judge whether α-b ∈ K5, or α - b ∈ K5, where α ∈ K5, b ∈ [0,1]. Here, we need to define a quinary expansion and an addition principle.Finally, we give a necessary and sufficient condition of α+FΦ (?) K5, and answer the following question:how can we judge α+FΦ (?) K5 for given a ∈ K5 and Φ ∈ F? Also, we characterize a special class of self-similar subsets α+FΦ of K5, where Φ includes the maps with equal contraction ratio.