Sierpiński Gasket as a Martin Boundary II —

  • Hiroshi Sato

    Kyushu University, Fukuoka, Japan
  • Manfred Denker

    Georg-August-Universität Göttingen, Germany


It is shown in [DS] that the Sierpiński gasket can be represented as the Martin boundary of a certain Markov chain and hence carries a canonical metric induced by the embedding into an associated Martin space . It is a natural question to compare this metric with the Euclidean metric. We show first that the harmonic measure coincides with the normalized -dimensional Hausdorff measure with respect to the Euclidean metric. Secondly, we define an intrinsic metric which is Lipschitz equivalent to and then show that is not Lipschitz equivalent to the Euclidean metric, but the Hausdorff dimension remains unchanged and the Hausdorff measure in is infinite. Finally, using the metric , 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 — . Publ. Res. Inst. Math. Sci. 35 (1999), no. 5, pp. 769–794

DOI 10.2977/PRIMS/1195143423