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

Abstract

It is shown in [DS] that the Sierpinski gasket ^aR N can be represented as the Martin boundary of a certain Markov chain and hence carries a canonical metric p M induced by the embedding into an associated Martin space M. It is a natural question to compare this metric p M with the Euclidean metric. We show first that the harmonic measure coincides with the normalized //=(log(Af+l)/log2)-dimensional Hausdorff measure with respect to the Euclidean metric. Secondly, we define an intrinsic metric p which is Lipschitz equivalent to p M 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.

On hyperbolic graphs induced by iterated function systems

“…Indeed motivated by the DS-type Markov chain (see [DS1,2,3], [JLW], [LW2], [RW], [DW]) that the sample paths go to the offsprings (vertical edges), and the offsprings of the neighbors (slanted edges), we can define a slanted set of edges on X (also on X ∼ as in Section 3) by E s = {(x, y) : |y| − |x| = 1, (x, y) ∈ E v , dist(K x , K y ) ≤ κr min{|x|,|y|} }; Note that in this case E v ∪ E s satisfies (i) there is no horizontal edges in the graph; and…”

“…Each element of K has a symbolic representation in Σ ∞ , i.e., there is a canonical surjection τ : Σ ∞ → K, and K is homeomorphic to the quotient space Σ ∞ / ∼, where the equivalence relation is defined by τ (x) = τ (y). In general one would like to impose more information on Σ * so as to carry out further analysis on K. With the intention to bring in the probabilistic potential theory to K, Denker and Sato [DS1,2,3] first constructed a special type of Markov chain {Z n } ∞ n=0 on Σ * of the Sierpinski gasket (SG), and showed that the Martin boundary of {Z n } ∞ n=0 is homeomorphic to the SG. Motivated by this, Kaimanovich [K] introduced the concept of "augmented tree" on Σ * by adding new edges to the tree Σ * according to the intersection of the cells of the IFS, he showed that the graph of the SG is hyperbolic in the sense of Gromov ([G], [Wo]), and that the SG is Hölder equivalent to the hyperbolic boundary of the augmented tree.…”

Hausdorff dimension for Martin metrics

“…A second definition can be formulated geometrically using equilateral similar triangles. Still another definition has been given in [1] in terms of transient Markov chains and its associated Martin boundary. The latter is defined as a completion of its state space with respect to certain metrics, which we call Martin metrics.…”

“…The Martin metrics have been investigated in [1], [2] and [7]. It has been shown in the first paper that the Martin boundary (with respect to the metric defined by r n = 2 −n ) is homeomorphic to the Sierpiński gasket.…”

“…It has been shown in the first paper that the Martin boundary (with respect to the metric defined by r n = 2 −n ) is homeomorphic to the Sierpiński gasket. In [2] it is also shown that this metric is not Lipschitz equivalent to the Euclidean metric; more precisely, the Hausdorff dimension is unchanged, but the h-dimensional Hausdorff measure is infinite. The Martin metrics are investigated in greater detail in [7].…”