Lipschitz equivalence of fractal triangles

Given an integer number n >= 2 and two digit sets A subset of {k(1)alpha + k(1)beta : k(1) + k(2) <= n - 1 and k(1), k(2) is an element of N boolean OR {0}}, 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), there is a self-similar set T = T(A, B) subset of R-2 satisfying the set equation: T = [T + A) boolean OR (B - T)]/n. We call such T a fractal triangle. By examining deeper the 'types' of connected components in each step of constructing such fractal triangles, we in this paper successfully characterize the Lipschitz equivalence of two classes of totally disconnected fractal triangles in it through the number of basic triangles in the first step of construction. (C) 2015 Elsevier Inc. All rights reserved.