Sierpienski垫片上Dirichlet型的递归构造
On a recursive construction of Dirichlet form on the Sierpiński gasket
摘要：
设$Γ_n$表示Sierpienski垫片$K$的$n$级Sierpienski图。对任意给定的$Γ_0$上的电导$(a_0, b_0, c_0)$，我们考虑由$Γ_n, n\geq 0$上的相容电导序列$(a_n, b_n, c_n)$的递归构造得到的$K$上的Dirchlet形式${\mathcal E}$.我们证明了存在一个二分法的情况：要么$a_0= b_0 =c_0$和${\mathcal E}$是标准Dirichlet型，要么是$a_0 >b_0 =c_0$（或两个对称的替代），而${\mathcal E}$是独立于$a_0, b_0$的非自相似Dirichlet型.第二种情况也已经在[Hattori et al 1994][Hambley et al 2002]中作为Sierpienski垫片上的一维渐近扩散过程进行了研究。对于谱性质，我们给出了相关Laplacian的特征值分布的精确估计，改进了[Hambley et al 2002]中的类似结果。
英文摘要：
Let $Γ_n$ 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 $Γ_0$, the Dirchlet form ${\mathcal E}$ on $K$ obtained from a recursive construction of compatible sequence of conductances $(a_n, b_n, c_n)$ on $Γ_n, n\geq 0$. 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 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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