Lipschitz equivalence of fractal sets in ℝ

Let T(q,D) be a self-similar (fractal) set generated by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$ \left\{ {fi(x) = \frac{1} {q}(x + d_i )} \right\}_{i = 1}^N $$ \end{document} where integer q > 1 and D = {d1, d2, …, dN} ⊂ ℝ. To show the Lipschitz equivalence of T(q,D) and a dust-like T(q,C), one general restriction is D ⊂ ℚ by Peres et al. [Israel J Math, 2000, 117: 353–379]. In this paper, we obtain several sufficient criterions for the Lipschitz equivalence of two self-similar sets by using dust-like graph-directed iterating function systems and combinatorial techniques. Several examples are given to illustrate our theory.