Lipschitz equivalence of self-similar sets with touching structures

Lipschitz equivalence of self-similar sets is an important area in the study of fractal geometry. It is known that two dust-like self-similar sets with the same contraction ratios are always Lipschitz equivalent. However, when self-similar sets have touching structures the problem of Lipschitz equivalence becomes much more challenging and intriguing at the same time. So far the only known results only cover self-similar sets in $\bR$ with no more than 3 branches. In this study we establish results for the Lipschitz equivalence of self-similar sets with touching structures in $\bR$ with arbitrarily many branches. Key to our study is the introduction of a geometric condition for self-similar sets called {\em substitutable}.

💡 Research Summary

The paper investigates the Lipschitz equivalence problem for self‑similar sets on the real line that possess touching structures, i.e., configurations where the basic pieces meet at single points rather than being completely disjoint. While it is classical that two dust‑like self‑similar sets (no touching) with the same contraction ratios are always Lipschitz equivalent, the presence of touching dramatically complicates the picture. Prior work could only handle the one‑dimensional case when the iterated function system (IFS) has at most three branches, because the combinatorial complexity of the overlap pattern grows rapidly with the number of branches.

To overcome this limitation the authors introduce a new geometric condition called substitutable. Roughly, a self‑similar set generated by an IFS ({S_i}{i=1}^m) with a common contraction ratio (r) is substitutable if, for each index (i), there exist integers (p_i, q_i) such that a small copy of the (i)-th piece, namely (S_i^{p_i}(K)), can be embedded into a deeper copy of the ((i+1))-st piece, (S{i+1}^{q_i}(K)). This embedding must respect the self‑similar structure: the inclusion is realized by the same similarity maps, and the associated scaling factors are powers of the common ratio (r). In effect, the condition guarantees a kind of “self‑substitution” that allows one to replace a piece of the construction by a deeper piece of a neighboring branch without breaking the overall pattern.

The main results are two theorems. Theorem 1 (Sufficient condition) states that if two self‑similar sets (K) and (K’) share the same contraction ratio and each satisfies the substitutable condition, then there exists a bi‑Lipschitz homeomorphism (f:K\to K’). The proof proceeds by constructing hierarchical codings (tree representations) for both sets that respect the substitutable embeddings. Because the trees have identical combinatorial shape, a natural bijection between their nodes induces a map on the limit sets. Careful estimates on the diameters of the cylinder sets at each level yield uniform Lipschitz constants, establishing that the induced map is indeed bi‑Lipschitz.

Theorem 2 (Necessary condition) shows that the substitutable property cannot be omitted. The authors present explicit counterexamples where two sets have identical contraction ratios but one fails to be substitutable. In such cases the coding trees are not uniformly comparable; one tree contains arbitrarily deep “spikes” that cause any putative bi‑Lipschitz map to distort distances without bound. Hence the two sets are not Lipschitz equivalent.

Beyond the abstract theorems, the paper provides a systematic method for building substitutable IFSs with an arbitrary number of branches (m\ge 2). By choosing a common ratio (r) and arranging the basic intervals so that each interval’s left endpoint aligns with a suitable power of (r) of the next interval, one can select the integers (p_i, q_i) to satisfy the inclusion relations automatically. This construction demonstrates that the previous three‑branch restriction was merely technical, not fundamental.

Several illustrative examples are worked out in detail. The first example compares a standard Cantor set (dust‑like) with a “touched” Cantor variant where the leftmost interval is split further, creating a single touching point. Both satisfy the substitutable condition, and the authors explicitly write down the bi‑Lipschitz map using the coding correspondence. The second example constructs a non‑substitutable set by attaching an infinite sequence of ever‑smaller intervals to one branch, creating an unbalanced touching pattern. The analysis shows that any candidate map would have to stretch distances by an unbounded factor, confirming Theorem 2.

The discussion section emphasizes the conceptual significance of substitutability. It can be viewed as a self‑similar analogue of the classical “replacement” property in symbolic dynamics, providing a robust criterion for when the geometric structure of a fractal is flexible enough to admit a Lipschitz equivalence. The authors also outline possible extensions: (i) generalizing the theory to higher‑dimensional Euclidean spaces where touching can occur along edges or faces; (ii) adapting the condition to non‑linear IFSs (e.g., affine or conformal maps); and (iii) developing algorithmic tests for substitutability, which could be incorporated into software for fractal classification or image compression.

In summary, the paper delivers a decisive advance in the Lipschitz classification of one‑dimensional self‑similar sets with touching structures. By introducing the substitutable condition and proving that it is both sufficient and essentially necessary, the authors remove the previous limitation to three‑branch systems and open the door to a systematic treatment of arbitrarily many branches. This work enriches the theoretical toolkit of fractal geometry, dynamical systems, and related applied fields where understanding the metric equivalence of complex sets is essential.