Lipschitz surjections between self-similar sets

Abstract: We give a characterization of the existence of Lipschitz surjections between dust-like self-similar sets of the same dimension under the assumption that the similarity ratios are commensurable. Our results apply more generally to graph-directed fractals.

• The paper characterizes the existence of Lipschitz surjections between dust-like self-similar sets, stating that a Lipschitz surjection $X → Y$ exists if and only if the union of similarity ratios $R(Φ) ∪ R(Ψ)$ is commensurable.
• An important corollary from the main theorem suggests that if either set of similarity ratios is commensurable, then the existence of a Lipschitz surjection in one direction ensures its existence in the reverse direction, even though Lipschitz surjections are not generally symmetric.
• The methodology involves a reduction to homogeneous primitive systems, addressing the general framework of graph-directed iterated function systems (GD-IFSs)

Overview and main result

This paper, by Attila Gáspár (2608.16471), studies the existence of Lipschitz surjections between dust-like self-similar sets of equal Hausdorff dimension. Recall that a compact set $X \subset \mathbb{R}^d$ is dust-like if it is generated by contracting similarities $S_1,\dots,S_n$ with pairwise disjoint pieces $S_i(X)$. Prior work established that a Lipschitz surjection $X \to Y$ exists whenever $\dim_H X > \overline{\dim}_B Y$ (Balka–Keleti), and that when $\dim_H X = \dim_H Y$, the existence of a Lipschitz surjection is equivalent to the existence of a bi-Lipschitz map provided either $X$ or $Y$ is homogeneous (all similarity ratios equal). The question of whether this equivalence persists without homogeneity was answered negatively by a counterexample of Ruan–Xiao.

The paper's main contribution is a characterization under a strictly weaker hypothesis: commensurability of similarity ratios. A set of ratios is commensurable if all ratios $\log r_1 / \log r_2$ are rational. The main theorem states:

Let $X$ and $S_1,\dots,S_n$0 be dust-like self-similar sets with IFSs $S_1,\dots,S_n$1, $S_1,\dots,S_n$2, both of dimension $S_1,\dots,S_n$3. If either $S_1,\dots,S_n$4 or $S_1,\dots,S_n$5 is commensurable, then there exists a Lipschitz surjection $S_1,\dots,S_n$6 if and only if $S_1,\dots,S_n$7 is commensurable.

A notable structural feature is the symmetry of the right-hand condition in $S_1,\dots,S_n$8 and $S_1,\dots,S_n$9. This yields a corollary that is arguably counterintuitive: if one of the two sets has commensurable ratios and they share dimension, then the existence of a Lipschitz surjection in one direction forces its existence in the reverse direction — even though Lipschitz surjections are not symmetric objects in general. The author explicitly leaves open whether this reversibility holds without any commensurability assumption.

The results are proved in the more general framework of graph-directed iterated function systems (GD-IFSs) in the sense of Mauldin–Williams: finite directed graphs whose edges carry contracting similarities, with attractors satisfying $S_i(X)$0. The paper assumes GD-IFSs are dust-like and strongly connected, which guarantees all attractors have common dimension $S_i(X)$1 with positive finite $S_i(X)$2-Hausdorff measure. In this setting, the role of individual ratios is played by cycle ratios $S_i(X)$3, and the theorem holds verbatim with $S_i(X)$4 the set of cycle ratios.

The graph-directed generality is not cosmetic: the proof genuinely requires it, since the key reduction (Proposition on powering homogeneous systems below) has no analogue for ordinary IFSs.

Reduction to homogeneous primitive systems

The first substantive ingredient shows that commensurability of cycle ratios lets one replace an arbitrary GD-IFS by a primitive one — homogeneous, strongly connected, with cycle-length GCD 1 — having the same attractor.

The argument combines three steps. First, a combinatorial potential lemma: if every directed cycle in a strongly connected graph has positive integer weight, then edge weights can be shifted by a potential so as to become nonnegative integers, positive on edges into a chosen base vertex. Applying this with weights $S_i(X)$5 (where $S_i(X)$6 generates the cycle-ratio group), rescaling each attractor by a power of $S_i(X)$7, and conjugating similarities yields a homogeneous system whose cycles have ratio less than 1 but which may contain isometric edges (ratio exactly 1). Isometries are eliminated by an edge-composition ("elimination") procedure that preserves strong connectivity, dust-likeness, the GCD invariant, and crucially cannot delete the base vertex because it was chosen to lack incoming isometric edges. Finally, edges of ratio $S_i(X)$8 are subdivided into paths of $S_i(X)$9 edges of ratio $X \to Y$0, introducing intermediate vertices whose attractors are scaled copies of existing ones.

Two further facts about primitive dust-like GD-IFSs are recorded: the system can be powered to any ratio $X \to Y$1 via length-$X \to Y$2 walk refinements (with adjacency matrix $X \to Y$3 still primitive); and if one component of the measure vector lies in the ring $X \to Y$4 where $X \to Y$5, then all components do. The latter follows from a linear-algebra argument: the Perron eigenspace of the integer adjacency matrix is one-dimensional, forcing the measure vector to coincide with a solution over $X \to Y$6, and $X \to Y$7 is algebraic so ring and field coincide.

Measure-linear surjections

The second section establishes, without any commensurability assumption, that a Lipschitz map $X \to Y$8 with positive-measure image gives rise to a Lipschitz surjection from a disjoint union of scaled copies of the $X \to Y$9 onto each $\dim_H X > \overline{\dim}_B Y$0, minimizing domain measure among such maps, whose pushforward is a constant multiple of Hausdorff measure. This generalizes Ruan–Xiao's work; notably, the resulting covering lemma (each small cell's image contains a cell of comparable diameter) is stronger than its IFS predecessor because homogeneity of $\dim_H X > \overline{\dim}_B Y$1 is not assumed.

Technically, the proof uses a martingale argument due to Wang in place of Ruan–Xiao's Proposition 3.4: the conditional measures of $\dim_H X > \overline{\dim}_B Y$2 on level-$\dim_H X > \overline{\dim}_B Y$3 cells form a bounded martingale, converging almost everywhere by Doob's theorem. Combined with the fact that almost every coding walk contains every cyclic walk (an exponential decay estimate using strong connectivity), one extracts a point $\dim_H X > \overline{\dim}_B Y$4 along which the preimage decomposition stabilizes, yielding boundedness of the minimal-domain function $\dim_H X > \overline{\dim}_B Y$5 on $\dim_H X > \overline{\dim}_B Y$6; a compactness/Arzelà–Ascoli limit argument then produces the surjective minimizer, and an equality-chasing argument over cells propagates the constant-density property across all vertices.

From surjections to commensurability

The forward implication splits into two cases depending on which system has commensurable ratios. Both exploit the same mechanism: the quantity $\dim_H X > \overline{\dim}_B Y$7 along a nested sequence of cells takes values in a finite set up to scaling, because the integer counting vectors $\dim_H X > \overline{\dim}_B Y$8 recording which level-$\dim_H X > \overline{\dim}_B Y$9 cells of $\dim_H X = \dim_H Y$0 map into $\dim_H X = \dim_H Y$1 are bounded independently of $\dim_H X = \dim_H Y$2. Consequently, changes of $\dim_H X = \dim_H Y$3 are either zero or uniformly bounded below (a discreteness phenomenon analogous to one used by Falconer–Marsh for bi-Lipschitz maps).

When $\dim_H X = \dim_H Y$4 is commensurable, choosing $\dim_H X = \dim_H Y$5 whose coding walk contains every cyclic walk $\dim_H X = \dim_H Y$6 of $\dim_H X = \dim_H Y$7 forces $\dim_H X = \dim_H Y$8 to be eventually constant along long stretches, and comparing values at two repeats of $\dim_H X = \dim_H Y$9 yields $X$0 — a rational power of $X$1. When instead $X$2 is commensurable, the argument runs along powers of a cycle $X$3 in the graph of $X$4: the covering lemma guarantees large image cells inside $X$5, minimality of the measure-linear surjection forces these preimages to lie inside $X$6 entirely, and finiteness of cell types up to similarity lets one compare two levels, again producing $X$7.

From commensurability to surjections

The reverse direction adapts Xi–Xiong's mass decomposition technique to the graph-directed setting. After reducing both systems to homogeneous primitive form and powering them to a common ratio $X$8 (using $X$9 from commensurability), the key lemma states: given a cell $Y$0 at some level and target measures $Y$1 drawn from a finite set within the ring $Y$2, for sufficiently deep refinement $Y$3 admits a partition into subcells realizing those exact measures.

The proof reduces to a purely arithmetic statement (essentially reused from Xi–Xiong) about splitting sequences of numbers from finite sets into two subsequences with equal sums, together with the identity $Y$4 for large $Y$5 when $Y$6 and $Y$7 ranges over nonnegative integer vectors, plus Perron–Frobenius convergence $Y$8 ensuring positivity of the partition vectors.

The surjection itself is built as a limit of cell-to-cell correspondences: starting from $Y$9 disjoint scaled copies of $\log r_1 / \log r_2$0 (whose existence at deep levels follows from strong connectivity plus a counting argument showing the attractor is not a single cycle of points), one recursively partitions preimage cells according to the measures of image cells using the mass decomposition lemma, maintaining the invariant $\log r_1 / \log r_2$1. Nested closed cells with vanishing diameters define a pointwise limit map, which is Lipschitz (separated points at depth $\log r_1 / \log r_2$2 sit at distance $\log r_1 / \log r_2$3 while images lie in cells of diameter $\log r_1 / \log r_2$4), has dense image by construction, hence is surjective by compactness. Extending constantly on the complement of the clopen set $\log r_1 / \log r_2$5 preserves Lipschitz continuity.

This construction also recovers, as a special case, a simpler proof of the earlier counterexample of Ruan–Xiao: two dust-like subsets of $\log r_1 / \log r_2$6, both of dimension $\log r_1 / \log r_2$7 with golden-ratio-related measure roots generating different rings $\log r_1 / \log r_2$8-incompatibly (hence not bi-Lipschitz equivalent by the Xi–Xiong criterion), yet admitting Lipschitz surjections in both directions since all ratios are powers of $\log r_1 / \log r_2$9.

Limitations and open questions

The characterization requires at least one of the two systems to have commensurable ratios (cycle ratios, in the graph-directed case), and dust-likeness throughout; the paper does not address self-similar sets with overlaps or with incommensurable ratios on both sides. The author explicitly states that he does not know whether the surjection-reversal corollary remains valid when neither $X$0 nor $X$1 has commensurable ratios. Additionally, the cited generalization of the bi-Lipschitz characterization to totally disconnected sets satisfying the open set condition involves substantially more complicated conditions, and no analogue of the present surjection characterization is claimed beyond the dust-like, strongly connected regime — where Mauldin–Williams theory ensures well-behaved Hausdorff measures; without strong connectivity the dimensional analysis becomes more delicate.

Conclusion

The paper replaces the homogeneity assumption in the Balka–Keleti/Ruan–Xiao equivalence between Lipschitz surjectivity and bi-Lipschitz equivalence with the weaker, symmetric condition of commensurable ratios, yielding a clean characterization of Lipschitz surjectivity between equal-dimensional dust-like self-similar sets and their graph-directed generalizations. The proof synthesizes a graph-theoretic reduction to primitive homogeneous systems, a martingale-based construction of measure-linear surjections, and a graph-directed adaptation of Xi–Xiong's mass decomposition method, and it supplies a transparent explanation of the known failure of the equivalence without commensurability.