As the investigation of the Lipschitz equivalence is very much restricted on a few special self-similar sets, the authors of this paper provide a broader framework to study the problem through the concept of an augmented tree introduced by V. A. Kaimanovich [in: P. Grabner (ed.) et al., Fractals in Graz 2001. Analysis, dynamics, geometry, stochastics. Proceedings of the conference, Graz, Austria, June 2001. Basel: Birkhäuser. Trends in Mathematics. 145–183 (2003; Zbl 1031.60033)]. K.-S. Lau and X.-Y. Wang [Indiana Univ. Math. J. 58, No. 4, 1777–1795 (2009; Zbl 1188.28005)] developed this concept further to general self-similar sets proving that, if an IFS satisfies the open set condition, then the augmented tree is hyperbolic in the sense of Gromov.
In the paper under review, it is shown that a simple augmented tree is always hyperbolic and the relationship of the hyperbolic boundary and the self-similar set is analogous to the case with the open set condition. The symbolic space, denoted by \(X\), has a natural graph structure and its vertical age set is denoted by \(\epsilon_V\) . Assuming that the IFS has an equal contraction ratio, the first basic theorem in the paper is: Suppose that the augmented tree \((X,\epsilon)\) is simple and the corresponding incidence matrix \(A\) is primitive, then \(\partial(X,\epsilon) \cong \partial(X,\epsilon_V)\). A less restrictive form of this theorem in the terms of rearrangeable matrices is proved also. The next basic theorem based on the previous theorem is concerned with a self-similar set called dust-like set: (i) if the IFS satisfies some mild conditions, then \(K\) is Lipschitz equivalent to a dust-like self-similar set; (ii) if \(K\) and \(K'\) are as in (i) and the two IFS’s have the same number of similitudes and the same contraction ratio, then they are Lipschitz equivalent. The main results are proved via several propositions, lemmas and theorems and illustrated with nice new examples. Some concluding remarks and open questions are also given.