LIPSCHITZ EQUIVALENCE OF TOTALLY DISCONNECTED GENERAL SIERPINSKI TRIANGLES

Given an integer n >= 2 and an ordered pair (A, B) with A subset of {k(1)alpha + k(2)beta : k(1) + k(2) <= n - 1 and k(1), k(2) is an element of N boolean OR {0}} and B subset of {k(1)alpha + k(2)beta : 2 <= k(1) + k(2) <= n and k(1), k(2) is an element of N}, where alpha = (1, 0), beta = (1/2, root 3/2). Let T := T (A, B) be unique compact set of R-2 satisfying the set equation: T = [(T + A) boolean OR (B - T)]/n. In this paper, we show that such self-similar sets which are totally disconnected are determined to within Lipschitz equivalence by their Hausdorff dimension.