Martin Metrics on the Sierpiński Gasket

It is known that the (N–1)-dimensional Sierpiński gasket [Formula: see text] is defined as an attractor of a dynamical system induced by affine mappings fi (i = 1, 2, …, N). Identifying fi with i, Denker and Sato [1] represented [Formula: see text] as the Martin boundary of a Markov chain on the word space generated by N letters. The Martin metrics determine a unique topological space on [Formula: see text] but are not always Lipschitz equivalent. In this paper, we characterize the Lipschitz equivalence of Martin metrics in terms of their coefficients, compare them with the Euclidean metric and compute their Hausdorff dimensions.