Weak Homogenization of Anisotropic Diiusion on Pre-sierpi Nski Carpets

We study a kind of`restoration of isotropy' on the pre-Sierpi nski carpet. Let R x n (r) and R y n (r) be the eeective resistances in the x and y directions, respectively, of the Sierpi nski 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 y 0 = R x 0. We prove that isotropy is weakly restored asymptotically in the sense that for all suuciently 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 eeective resistances asymptotically grow exponentially with an exponent equal to that found by Barlow and Bass for the isotropic case r = 1.