Self-similarity Of Union Of Self-similar Set And Its Translation

Let an iterated function system be {f₁(x)=x/5,f₂(x)=x/5+2/5,f₁(x)=x/5+4/5},then there exists a self-similar set M which satisfiesM=(?)f_i(M)=1/5-M∪1/5-(M+2)∪1/5(M+4).In this paper, we mainly study the problem that, in which condition the a is , the union of the self-similar set M and its translation M+α={x+α:x∈M} can be self-similar. The main result is as follows:Theorem The union of the self-similar set M and its translation M+αis a self-similar set if and only if there exists a natural number n such thatα=±2.5ⁿ.