On a recursive construction of Dirichlet form on the Sierpi\'nski gasket

Let ${\mathrm{\Gamma }}_n$ denote the $n$-th level Sierpi'nski graph of the Sierpi'nski gasket $K$. We consider, for any given conductance $(a_0,b_0,c_0)$ on ${\mathrm{\Gamma }}_0$, the Dirchlet form $\mathcal{E}$ on $K$ obtained from a recursive construction of compatible sequence of conductances $(a_n,b_n,c_n)$ on ${\mathrm{\Gamma }}_n,n\ge 0$. We prove that there is a dichotomy situation: either $a_0= b_0 =c_0$and${\mathcal E}$isthestandardDirichletform,or$a_0 >b_0 =c_0$(orthetwosymmetricalternatives),and${\mathcal E}$ is a non-self-similar Dirichlet form independent of $a_0,b_0$. The second situation has also been studied in [Hattori et al 1994][Hambley et al 2002] as a one-dimensional asymptotic diffusion process on the Sierpi'nski gasket. For the spectral property, we give a sharp estimate of the eigenvalue distribution of the associated Laplacian, which improves a similar result in [Hambley et al 2002].