On a recursive construction of Dirichlet form on the Sierpiński gasket

2017/07/05 by Gu, Qingsong, Lau, Ka-Sing, Qiu, Hua
#28A80 #46E30 #46E35 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1707.01426

Abstract

Let Γ_n denote the n-th level Sierpiński graph of the Sierpiński gasket K. We consider, for any given conductance (a₀, b₀, c₀) on Γ₀, the Dirchlet form \mathcal E on K obtained from a recursive construction of compatible sequence of conductances (a_n, b_n, c_n) on Γ_n, n≥ 0. We prove that there is a dichotomy situation: either a₀= b₀ =c₀ and \mathcal E is the standard Dirichlet form, or a₀ >b₀ =c₀ (or the two symmetric alternatives), and \mathcal E is a non-self-similar Dirichlet form independent of a₀, b₀. 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].

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