It is shown in [DS] that the Sierpinski gasket \mathscr{S}⊂
RN can be represented as the Martin boundary of a certain Markov chain and hence carries a canonical metric ρ
M induced by the embedding into an associated Martin space
M. It is a natural question to compare this metric ρ
M with the Euclidean metric. We show first that the harmonic measure coincides with the normalized
H=(log(
N+1)/log2)-dimensional Hausdorff measure with respect to the Euclidean metric. Secondly, we define an intrinsic metric ρ which is Lipschitz equivalent to ρ
M and then show that ρ is not Lipschitz equivalent to the Euclidean metric, but the Hausdorff dimension remains unchanged and the Hausdorff measure in ρ is infinite. Finally, using the metric ρ, we prove that the harmonic extension of a continuous boundary function converges to the boundary value at every boundary point.
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