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

### 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 *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.