Weak homogenization of anisotropic diffusion on pre-Sierpiński carpets

We study a kind of 'restoration of isotropy" on the pre-Sierpiński carpet. Let R^x_n(r) and R^y_n(r) be the effective resistances in the x and y directions, respectively, of the Sierpiński carpet at the n^th stage of its construction, if it is made of anisotropic material whose anisotropy is parametrized by the ratio of resistances for a unit square: r = R^y₀ / R^x₀. We prove that isotropy is weakly restored asymptotically in the sense that for all sufficiently large n the ratio R^y_n(r) / R^x_n(r) is bounded by positive constants independent of r. The ratio decays exponentially fast when r ≫ 1. Furthermore, it is proved that the effective resistances asymptotically grow exponentially with an exponent equal to that found by Barlow and Bass for the isotropic case r = 1.