On the Hausdorff Dimension of the Sierpiński Gasket with respect to the Harmonic Metric

The spectral dimension and the Hausdorff dimension of the Sierpiński gasket are known to be different. Kigami proved that the Hausdorff dimension dh with respect to the harmonic metric is less than or equal to the spectral dimension and conjectured equality. We give estimates of dh which are strictly less than the spectral dimension and therefore disprove this conjecture. 1. Sierpiński gasket and Sierpiński graphs The Sierpiński gasket G is a well known selfsimilar set introduced by Sierpiński as a curve all of whose points are ramification points. For i = 1, 2, 3 we define the maps φi(x) = ai + 1 2 (x− ai), where a1 = (0, 0), a2 = (1, 0) and a3 = 1 2 (1, √ 3). Then G is the invariant set with respect to the similarities φ1, φ2 and φ3, i. e. G = φ1(G) ∪ φ2(G) ∪ φ3(G). Figure 1: The Sierpiński gasket. The Sierpiński graphs Γn = (Vn, En) are defined inductively in the following way: For n = 0 we set V0 = {a1, a2, a3} and E0 = {a1a2, a1a3, a2a3}. If n > 0 let Vn = φ1(Vn−1) ∪ φ2(Vn−1) ∪ φ3(Vn−1) and the edges are inherited from the previous stage. For n ∈ N let Wn = {1, 2, 3}n be the word space consisting of all words over {1, 2, 3} of length n and let W = {1, 2, 3}N. Moreover we write φw = φw1 ◦· · ·◦φwn for the composition of similarities φw1 , . . . , φwn for a finite word w = w1 . . . wn. Then we can define a map π from W onto G by