Sierpiński Gasket as a Martin Boundary II ($\text{The Intrinsic Metric}$)

Abstract

It is shown in [DS] that the Sierpiński gasket $\mathcal{S}\in {\mathbf{R}}^N$ can be represented as the Martin boundary of a certain Markov chain and hence carries a canonical metric ${\rho }_M$ induced by the embedding into an associated Martin space $M$. It is a natural question to compare this metric ${\rho }_M$ with the Euclidean metric. We show first that the harmonic measure coincides with the normalized $H=(\log (N+1)/\log 2)$-dimensional Hausdorff measure with respect to the Euclidean metric. Secondly, we define an intrinsic metric $\rho$ which is Lipschitz equivalent to ${\rho }_M$ and then show that $\rho$ is not Lipschitz equivalent to the Euclidean metric, but the Hausdorff dimension remains unchanged and the Hausdorff measure in $\rho$ is infinite. Finally, using the metric $\rho$, we prove that the harmonic extension of a continuous boundary function converges to the boundary value at every boundary point.