Sierpiński Gasket as a Martin Boundary II — (<em>The Intrinsic Metric</em>)

Abstract

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