On a dendrite generated by a zero-dimensional weak self-similar set

Let S be a zero-dimensional, perfect, compact weak self-similar set generated in dendrite X by a family {fj} of weak contractions from X to itself. Decomposition space Df of S due to a continuous mapping f from S onto X is also a dendrite. In the dendrite Df, there exists a zero-dimensional, perfect, compact weak self-similar set S1 based on a family {fj1} each of which is topologically conjugate to fj. Decomposition space Df1Df1 of S1 due to a continuous mapping f1 from S1 onto Df is again a dendrite. In this manner, through the successive formation of weak self-similar set, we can obtain a sequence X,Df,Df1,…X,Df,Df1,… of dendrite any pair in which are mutually homeomorphic.