On a recursive construction of Dirichlet form on the Sierpiński gasket

Abstract

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 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 a 0 = b 0 = c 0 and E is the standard Dirichlet form, or a 0 > b 0 = c 0 (or the two symmetric alternatives), and E is a non-self-similar Dirichlet form independent of a 0 , b 0 . The second situation has also been studied in [9, 10] 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 [10].

Dirichlet forms and critical exponents on fractals

“…This diffusion was investigated further by Hambly and Kumagai [12] on some other nested fractals (see also [10,14]). Recently, in [8], the authors gave a detail study of this method on the Sierpinski gasket from an analytic point of view; they showed a dichotomy result that for any initial data, the Dirichlet forms obtained are either the standard or the one in [15]. They also obtained sharp estimate of the eigenvalue counting functions of the associated Laplacian with respect to the α-Hausdorff measure.…”

“…The other construction, we call it reverse recursive method, is to fix an initial data at V 0 , and iterate this to V n to obtain a sequence of compatible networks. This method first appeared in a probabilistic study by Hattori, Hattori and Watanabe [15] on the Sierpinski gasket K (abc-gasket), they showed that there is an asymptotically one-dimensional diffusion process on K. Some further development and extensions can be found in [10,11,12,14] by Hambly et al, and in [8] by the authors.…”