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

TL;DR

This paper investigates a recursive method to construct Dirichlet forms on the Sierpiński gasket, revealing a dichotomy in the forms' self-similarity and providing sharp spectral estimates for the associated Laplacian.

Contribution

It establishes a clear dichotomy in the recursive construction of Dirichlet forms on the Sierpiński gasket and improves spectral eigenvalue estimates for the Laplacian.

Findings

Identifies conditions for standard and non-self-similar Dirichlet forms.

Provides sharp eigenvalue distribution estimates for the Laplacian.

Clarifies the independence of non-self-similar forms from initial conductances.

Abstract

Let $\Gamma_n$ 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 $\Gamma_0$, the Dirchlet form ${\mathcal E}$ on $K$ obtained from a recursive construction of compatible sequence of conductances $(a_n, b_n, c_n)$ on $\Gamma_n, n\geq 0$. 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 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].