# Sierpiński Gasket as a Martin Boundary II — $(The Intrinsic Metric)$

### Hiroshi Sato

Kyushu University, Fukuoka, Japan### Manfred Denker

Georg-August-Universität Göttingen, Germany

## 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 $ρ_{M}$ 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 $H=(g(N+1)/log2)$-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.

## Cite this article

Hiroshi Sato, Manfred Denker, Sierpiński Gasket as a Martin Boundary II — $(The Intrinsic Metric)$. Publ. Res. Inst. Math. Sci. 35 (1999), no. 5, pp. 769–794

DOI 10.2977/PRIMS/1195143423