The Moran--Hutchinson formula in semimetric spaces

Abstract: We establish the Moran--Hutchinson formula for attractors of finite systems of surjective similitudes on semimetric spaces. More precisely, for a complete, normal semimetric space satisfying strong regularity and geometric doubling, we prove that the open set condition implies that the Hausdorff measure of the attractor at the similarity dimension is positive and finite. We also prove the converse, specifically, positivity of the Hausdorff measure at the similarity dimension implies the open set condition. This provides a partial answer to a question posed by Bessenyei and Pénzes in 2022.

• The paper proves that, under completeness, strong regularity, geometric doubling, and the OSC, surjective similitude systems satisfy 0 < H^α(K) < ∞ and dim_H K = dim_M K = α.
• The paper constructs an invariant measure and adapts mass-distribution, packing, open-cover, and hyperspace methods to overcome non-open balls and the absence of the triangle inequality.
• The paper establishes that positive α-dimensional Hausdorff measure forces the OSC, while identifying open problems involving uniqueness of invariant measures, removal of surjectivity, and weaker regularity or doubling assumptions.

The Moran–Hutchinson formula is a cornerstone of fractal geometry: for a finite iterated function system (IFS) of contracting similitudes satisfying the open set condition (OSC), the Hausdorff measure of the attractor at the similarity dimension is positive and finite, and the Hausdorff dimension equals the similarity dimension. Schief established a converse in Euclidean space, showing that positivity of this Hausdorff measure forces the OSC. The paper under review, "The Moran–Hutchinson formula in semimetric spaces" (2608.16817), extends both directions to semimetric spaces — spaces where symmetry and separation hold but the triangle inequality may fail — thereby giving a partial answer to a question posed by Bessenyei and Penzes.

Background and motivation

Hutchinson proved that every finite family of contractions on a complete metric space has a unique compact attractor, and that for similitudes on Euclidean space satisfying the OSC, the attractor's Hausdorff dimension equals its similarity dimension with positive and finite Hausdorff measure there. In arbitrary complete metric spaces, however, the OSC alone does not suffice; Schief exhibited counterexamples, and Balogh and Rohner restored the theorem under a doubling assumption. Rajala and Vilppolainen generalized the doubling-space theory further, while their non-bijective similitude example shows that the OSC alone need not determine the Hausdorff dimension from the contraction ratios even in metric spaces.

Semimetric spaces offer a genuinely different extension. A semimetric $d$ satisfies $d(x,y)=0$ iff $x=y$ and symmetry, but not the triangle inequality; balls need not be open sets. To recover usable structure, Bessenyei and Pales introduced a regularity condition (a local substitute for the triangle inequality), and Bessenyei and Penzes introduced normality; together these guarantee existence and uniqueness of attractors for weak contractions. The present paper adds two further hypotheses — strong regularity and geometric doubling — and proves the full Moran–Hutchinson dichotomy for surjective similitudes.

Framework

A triangle function $\Phi$ satisfies $d(x,z) \le \Phi(d(x,y), d(y,z))$; the canonical choice is the basic triangle function $\Phi_d$, defined as the supremum of $d(x,y)$ over pairs $(x,y)$ both within distances $u$ and $v$ of some common point. The space is regular if $d(x,y)=0$0 is continuous at $d(x,y)=0$1 and normal if $d(x,y)=0$2 for finite arguments. Regularity ensures that interiors are open, that topological and sequential closures coincide, and that compactness coincides with sequential compactness; normality ensures boundedness is equivalent to finite diameter. Under regularity and completeness, closed totally bounded sets are compact, and Kocsis–Pales / Bessenyei–Penzes guarantee a unique attractor $d(x,y)=0$3 for finite families of $d(x,y)=0$4-contractions with right-continuous comparison functions. Quasi-metric ($d(x,y)=0$5-metric) spaces are subsumed as they are automatically normal and regular.

Two additional assumptions drive the main results:

• Strong regularity: $d(x,y)=0$6, i.e., the triangle defect is locally Lipschitz at scale zero.
• Geometric doubling: every ball of radius $d(x,y)=0$7 is covered by at most $d(x,y)=0$8 balls of radius $d(x,y)=0$9. The author notes that a local version suffices where doubling is used.

Both hold automatically in quasi-metric spaces.

Invariant measure

For weights $x=y$0 summing to one, the paper constructs a Borel probability measure $x=y$1 supported on the attractor satisfying $x=y$2. The construction follows the standard coding map route: the natural projection $x=y$3 is shown to be a continuous surjection using right-continuity of the comparison functions and the decay $x=y$4, and $x=y$5 is defined as the push-forward of the Bernoulli product measure. Notably, uniqueness of the invariant measure is not claimed — a genuine gap relative to the metric-space theory, where uniqueness follows from weak contraction arguments that do not transfer directly here.

The Moran–Hutchinson formula

The main theorem states: if $x=y$6 is complete, regular, normal, strongly regular, and geometrically doubling, and $x=y$7 is a finite family of surjective similitudes with ratios $x=y$8 satisfying the OSC, then for $x=y$9 with $\Phi$0,

$\Phi$1

The proof adapts the mass distribution argument of Bishop–Peres, but requires substantial modification because balls may fail to be open or even Borel. Three technical devices carry the argument:

1. A packing-type lemma showing that any family of disjoint sets, each containing a ball of radius $\Phi$2 and contained in a ball of radius $\Phi$3 and meeting a set of diameter at most $\Phi$4, has cardinality bounded by $\Phi$5 uniformly in small $\Phi$6.
2. A mass distribution bound $\Phi$7 for open sets $\Phi$8 of diameter at most $\Phi$9, obtained by decomposing along the minimal cut-set $d(x,z) \le \Phi(d(x,y), d(y,z))$0 and counting, via the disjointness lemma, how many cylinder images $d(x,z) \le \Phi(d(x,y), d(y,z))$1 can meet $d(x,z) \le \Phi(d(x,y), d(y,z))$2.
3. Non-atomicity of $d(x,z) \le \Phi(d(x,y), d(y,z))$3, which handles degenerate covering pieces of diameter zero — necessary because singletons contribute nothing to the $d(x,z) \le \Phi(d(x,y), d(y,z))$4-dimensional cost but might otherwise appear in covers.

A further lemma circumvents the possible non-openness of balls: any set of small positive diameter is contained in an open set whose diameter exceeds it by at most a fixed factor $d(x,z) \le \Phi(d(x,y), d(y,z))$5. This yields the lower bound $d(x,z) \le \Phi(d(x,y), d(y,z))$6; finiteness and the Minkowski dimension bound follow as in the classical argument. An accompanying remark gives a partial converse: if $d(x,z) \le \Phi(d(x,y), d(y,z))$7 for some $d(x,z) \le \Phi(d(x,y), d(y,z))$8, then $d(x,z) \le \Phi(d(x,y), d(y,z))$9, with equality when pairwise intersections have vanishing $\Phi_d$0-measure — correcting in passing a missing surjectivity hypothesis in [BessenyeiPenzes2022, Theorem 3].

Necessity of the open set condition

The second main theorem extends Schief's converse: under strong regularity, geometric doubling, and surjective similitudes, positivity of $\Phi_d$1 implies the OSC. The proof follows the Bishop–Peres simplification of Schief's argument, with departures forced by the semimetric structure. Key steps include:

• Introduction of an open-cover variant $\Phi_d$2 of Hausdorff measure, needed because arbitrary covers cannot be related to open ones without the triangle inequality; all Borel sets are $\Phi_d$3-measurable, and $\Phi_d$4 holds via rescaling by compositions of the similitudes.
• A Hausdorff-distance separation lemma: incomparable words $\Phi_d$5 with $\Phi_d$6 satisfy $\Phi_d$7, where $\Phi_d$8 is a positive separation distance between $\Phi_d$9 and a suitable open neighborhood's complement; its proof uses additivity of $d(x,y)$0 on incomparable cylinders.
• Compactness of closed balls (from geometric doubling plus completeness), total boundedness of the hyperspace $d(x,y)$1 of closed subsets of a compactum, and a uniform bound on the size of the families $d(x,y)$2 of cut-set elements meeting $d(x,y)$3.

From these, the author extracts a distinguished word $d(x,y)$4 maximizing $d(x,y)$5, builds the open set $d(x,y)$6 with $d(x,y)$7 a union of suitably shrunk interiors of balls around $d(x,y)$8, and verifies $d(x,y)$9 and pairwise disjointness of the $(x,y)$0 using the set-distance separation estimate. A structural difficulty specific to semimetric spaces is that the diameter of an $(x,y)$1-neighborhood $(x,y)$2 of $(x,y)$3 cannot be controlled as $(x,y)$4; consequently the cut-set definition must be scaled by $(x,y)$5 rather than normalized, and no normalization $(x,y)$6 is imposed.

Examples

Two concrete classes illustrate applicability. First, on $(x,y)$7 with $(x,y)$8 for arbitrary $(x,y)$9 (a genuine semimetric when $u$0), the two-map IFS $u$1, $u$2 consists of surjective similitudes. When $u$3 the OSC holds and the theorem yields $u$4 with $u$5; when the sum exceeds $u$6, the attractor is $u$7 with $u$8, so the formula fails precisely when the OSC fails — consistent with the converse theorem. Second, anisotropic products on $u$9 with $v$0 admit four-map corner systems to which the theorem applies whenever the four image rectangles are pairwise disjoint.

Limitations and open questions

Several restrictions delimit the scope of the results. The maps must be surjective similitudes; Rajala–Vilppolainen treat a strictly wider class of contractions in the metric setting, so the present results neither contain nor directly generalize the doubling-metric theory of Balogh–Rohner or Rajala–Vilppolainen — the extensions proceed along different axes. Uniqueness of the invariant measure is left unproven, and the strong regularity and geometric doubling assumptions, while satisfied by quasi-metric spaces, are additional hypotheses not required for attractor existence itself. Whether the OSC alone (without strong regularity or doubling) suffices in semimetric spaces remains open, as does the extension beyond surjective similitudes; the failure mode in the one-dimensional example shows the converse direction is sharp at least there.

Conclusion

The paper establishes the Moran–Hutchinson formula, in both sufficiency and necessity directions, for finite systems of surjective similitudes on complete, regular, normal, strongly regular, geometrically doubling semimetric spaces. The proofs require genuine innovations — an open-cover Hausdorff measure, diameter-controlled open enlargements of arbitrary sets, and hyperspace compactness arguments — to compensate for the absence of the triangle inequality and the possible non-openness of balls. Together with the worked examples on power-type semimetrics, the results substantially broaden the class of spaces on which the classical dimension theory of self-similar sets is valid, while leaving the removal of the surjectivity and doubling-type hypotheses as concrete open problems.