Convergence of resistances on generalized Sierpinski carpets

The locally symmetric diffusions, also known as Brownian motions, on generalized Sierpinski carpets were constructed by Barlow and Bass in 1989. On a fixed carpet, by the uniqueness theorem (Barlow-Bass-Kumagai-Teplyaev, 2010), the reflected Brownians motion on level $n$ approximation Euclidean domain, running at speed $\lambda_n\asymp \eta^n$ with $\eta$ being a constant depending on the fractal, converges weakly to the Brownian motion on the Sierpinski carpet as $n$ tends to infinity. In this talk, we show the convergence of $\lambda_n/\eta^n$. We also give a positive answer to a closely related open question of Barlow-Bass (1990) about the convergence of the renormalized effective resistances between two opposite faces of approximation domains. This talk is based on a joint work with Zhen-Qing Chen.