On a recursive construction of Dirichlet form on the Sierpiński gasket

Abstract

Let ${\Gamma }_n$ denote the $n$-th level Sierpiński graph of the Sierpiński gasket $K$. We consider, for any given conductance $(a_0,b_0,c_0)$ on ${\Gamma }_0$, the Dirchlet form $\mathcal{E}$ on $K$ obtained from a recursive construction of compatible sequence of conductances $(a_n,b_n,c_n)$ on ${\Gamma }_n,n\ge 0$. We prove that there is a dichotomy situation: either $a_0=b_0=c_0$ and $\mathcal{E}$ is the standard Dirichlet form, or $a_0>b_0=c_0$ (or the two symmetric alternatives), 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 Sierpiński 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].

20 pages, 7 figures