Sierpi Nski Gasket as a Martin Boundary Ii (the Intrinsic Metric)

It is shown in DS] that the Sierpi nski gasket S IR N can be represented as the Martin boundary of a certain Markov chain and hence carries a canonical metric M induced by the embedding into an associated Martin space M. It is a natural question to compare this metric M with the Euclidean metric. We show rst that the harmonic measure coincides with the normalized H = (log(N + 1)= log 2)-dimensional Hausdorr measure with respect to the Euclidean metric. Secondly, we deene an intrinsic metric which is Lipschitz equivalent to M and then show that is not Lipschitz equivalent to the Euclidean metric, but the Hausdorr dimension remains unchanged and the Hausdorr measure in is innnite. Finally, using the metric , we prove that the harmonic extension of a continuous boundary function converges to the boundary value at every boundary point.