The Existence Of The Harmonic Structure Of Hata's Tree-like Set

Let X=C, f1(z)= cz, f2(z)= (1-│c│2)z+│c│2 where|c|,|1-c│∈(0,1). The self-similar set with respect to {f1,f2} is called Hata's tree-like set.This paper mainly studies the similar dimension of Hata's tree-like set and the existence of its harmonic structure.First of all, based on the definition and properties of a Hata's tree-like set, the similar dimension of a Hata's tree-like set are given by heuristic thinking. Secondly, Let r= (r1,r2),(0< r1,r2< 1), we use two different ideas to give a sufficient condition for (D, r) is regular harmonic structure. Moreover, we find that the sufficient condition has nothing to do with the initial values on the vertices V0.At last, we draw the variation trend of r1 and r2 with respect to the pa-rameter h, and then obtain the relationship between them.