The Existence And Uniqueness Of The Harmonic Structure On Three Types Of Sierpinski Gaskets

In this paper,we study the existence and uniqueness of the harmonic structure on finitely ramified self-similar fractals.For any given renormalization coefficients r:{ri,i = 1,2,…,s}corresponding to an iterated function system {fi,i= 1,2,…,5},we claim that on what condition should r = {ri,i 1,2,…,5} be satisfied to guarantee the existence of harmonic structure on the self-similar set K generated by the iterated function system.Moreover,if it exists,is it unique?We focus on three typical finitely ramified self-similar fractals,which are the standard Sierpinski gasket,Sierpinski gasket with twist and the tetrahedral Sierpinski gasket.We discuss the existence and uniqueness of the harmonic structure on these typical examples.We will give the condition that r =(ri,i = 1,2,…,5} should be satisfied to guarantee the existence and uniqueness of harmonic structure on the standard Sierpinski gasket and Sierpinski gasket with twist.For the tetrahedral Sierpinski gasket,we will introduce a special symmetry to find the condition for r = {fi,i = 1,2,…,s} to guarantee the existence and uniqueness of harmonic structure.