Fractal Quasi-Equivalence Principle

• The fractal quasi-equivalence principle is a framework that asserts small, controlled perturbations in fractal systems leave key invariants, such as Hausdorff dimension and critical measures, essentially unchanged.
• It employs modified Hausdorff metrics and quasi-unitary equivalences to compare shapes and energy forms, thereby extending classical results like Moran’s theorem to perturbed fractal structures.
• Applications span geometrical fractals, operator theory on fractals, and symbolic expansions in number theory, with thresholds like δ < 1 marking the boundary for preserving fractal properties.

Searching arXiv for the cited works and closely related uses of “fractal quasi-equivalence principle.” The fractal quasi-equivalence principle denotes a class of preservation statements asserting that controlled perturbations or structured correspondences between fractal models leave central invariants essentially unchanged. In its geometric formulation, due to Junyang Yu, the principle states that small perturbations of the defining similitudes of a self-similar set produce a nearby invariant set in a Hausdorff-type shape metric while preserving the Moran dimension and the positivity and finiteness of the critical Hausdorff measure under the open set condition (Yu, 2009). Related formulations appear in operator theory, where quasi-unitary equivalence compares energy forms on fractals, graphs, metric graphs, and graph-like manifolds (Post et al., 2018), and in metric number theory, where digit-shifting maps or transference constructions preserve Hausdorff dimension for sets defined by Perron, Engel, Pierce, or continued-fraction expansions under explicit hypotheses (Moroz, 3 Oct 2025, Nakajima et al., 15 Feb 2025). This suggests that “quasi-equivalence” is best understood not as a single theorem but as a recurrent structural theme: perturb the presentation, preserve the fractal content.

1. Metric framework and shape comparison

Yu’s formulation begins with several modifications of the Hausdorff metric for nonempty compact subsets of a metric space, especially subsets of $\mathbb{R}^n$. For compact sets $A,B\subset X$, the usual Hausdorff distance is

$$h(A,B)=\max\Big\{\sup_{a\in A}d(a,B),\;\sup_{b\in B}d(b,A)\Big\}.$$

To compare sets up to ambient symmetries rather than pointwise placement, Yu defines the absolute shape difference

$$\hat h(A,B)=\inf\bigl\{\,h(\varphi(A),\psi(B)):\;\varphi,\psi\ {\rm are\ isometries of }\mathbb{R}^n\}.$$

If $\tilde A$ denotes the isometry class of $A$, then $\hat h$ is a complete metric on compact subsets of $\mathbb{R}^n$ modulo isometry. Restricting $\varphi,\psi$ to orientation-preserving isometries yields the absolute rigid-shape difference $\hat h_{\rm rigid}$. For bounded sets with positive radii

$A,B\subset X$0

Yu normalizes by scale and defines the relative shape difference

$A,B\subset X$1

which descends to a complete metric on compact sets up to similarity. Combining radius normalization with orientation-preserving similarities gives the relative rigid-shape difference $A,B\subset X$2; it is again complete (Yu, 2009).

These metrics shift attention from exact coincidence to equivalence of form. In Yu’s terminology, they provide a way to “gaze on fractals in an aspect of their ‘form’,” and they support the later definitions of $A,B\subset X$3-quasi-self-similarity, quasi-prototiles, and quasi-motifs.

2. $A,B\subset X$4-quasi-self-similar sets

Fix an iterated-function-system of similitudes $A,B\subset X$5, where each $A,B\subset X$6 has Lipschitz ratio $A,B\subset X$7. Its invariant self-similar set $A,B\subset X$8 satisfies

$A,B\subset X$9

For a multi-index $$h(A,B)=\max\Big\{\sup_{a\in A}d(a,B),\;\sup_{b\in B}d(b,A)\Big\}.$$0, Yu writes

$$h(A,B)=\max\Big\{\sup_{a\in A}d(a,B),\;\sup_{b\in B}d(b,A)\Big\}.$$1

A perturbation is described by a structure system $$h(A,B)=\max\Big\{\sup_{a\in A}d(a,B),\;\sup_{b\in B}d(b,A)\Big\}.$$2 for a compact set $$h(A,B)=\max\Big\{\sup_{a\in A}d(a,B),\;\sup_{b\in B}d(b,A)\Big\}.$$3, indexed by finite words, such that

$$h(A,B)=\max\Big\{\sup_{a\in A}d(a,B),\;\sup_{b\in B}d(b,A)\Big\}.$$4

Given nonnegative errors $$h(A,B)=\max\Big\{\sup_{a\in A}d(a,B),\;\sup_{b\in B}d(b,A)\Big\}.$$5 with $$h(A,B)=\max\Big\{\sup_{a\in A}d(a,B),\;\sup_{b\in B}d(b,A)\Big\}.$$6, Yu calls $$h(A,B)=\max\Big\{\sup_{a\in A}d(a,B),\;\sup_{b\in B}d(b,A)\Big\}.$$7 a $$h(A,B)=\max\Big\{\sup_{a\in A}d(a,B),\;\sup_{b\in B}d(b,A)\Big\}.$$8-perturbation of $$h(A,B)=\max\Big\{\sup_{a\in A}d(a,B),\;\sup_{b\in B}d(b,A)\Big\}.$$9, or $$\hat h(A,B)=\inf\bigl\{\,h(\varphi(A),\psi(B)):\;\varphi,\psi\ {\rm are\ isometries of }\mathbb{R}^n\}.$$0-quasi-self-similar, if

$$\hat h(A,B)=\inf\bigl\{\,h(\varphi(A),\psi(B)):\;\varphi,\psi\ {\rm are\ isometries of }\mathbb{R}^n\}.$$1

When $$\hat h(A,B)=\inf\bigl\{\,h(\varphi(A),\psi(B)):\;\varphi,\psi\ {\rm are\ isometries of }\mathbb{R}^n\}.$$2, one speaks of a $$\hat h(A,B)=\inf\bigl\{\,h(\varphi(A),\psi(B)):\;\varphi,\psi\ {\rm are\ isometries of }\mathbb{R}^n\}.$$3-quasi-self-similar set.

The dimensional theorem requires an Open Set Condition formulated directly for the perturbed structure system. There must exist disjoint open sets $$\hat h(A,B)=\inf\bigl\{\,h(\varphi(A),\psi(B)):\;\varphi,\psi\ {\rm are\ isometries of }\mathbb{R}^n\}.$$4 such that: (i) $$\hat h(A,B)=\inf\bigl\{\,h(\varphi(A),\psi(B)):\;\varphi,\psi\ {\rm are\ isometries of }\mathbb{R}^n\}.$$5; (ii) $$\hat h(A,B)=\inf\bigl\{\,h(\varphi(A),\psi(B)):\;\varphi,\psi\ {\rm are\ isometries of }\mathbb{R}^n\}.$$6 for $$\hat h(A,B)=\inf\bigl\{\,h(\varphi(A),\psi(B)):\;\varphi,\psi\ {\rm are\ isometries of }\mathbb{R}^n\}.$$7; (iii) each $$\hat h(A,B)=\inf\bigl\{\,h(\varphi(A),\psi(B)):\;\varphi,\psi\ {\rm are\ isometries of }\mathbb{R}^n\}.$$8 contains a ball of radius $$\hat h(A,B)=\inf\bigl\{\,h(\varphi(A),\psi(B)):\;\varphi,\psi\ {\rm are\ isometries of }\mathbb{R}^n\}.$$9 and lies in a ball of radius $\tilde A$0 (Yu, 2009).

This formulation is stricter than a vague “small perturbation” heuristic. The perturbation is controlled at every cylinder level, scaled by the contraction factor $\tilde A$1, and measured against the radius $\tilde A$2 of the unperturbed attractor.

3. Moran-type preservation, examples, and failure modes

Yu’s main theorem is a perturbative extension of Moran’s classical theorem. Let $\tilde A$3 be the unique solution of

$\tilde A$4

If $\tilde A$5 is a $\tilde A$6-quasi-self-similar set satisfying the Open Set Condition, then

$\tilde A$7

(Yu, 2009).

The proof follows the classical Moran scheme but with perturbed cylinder sets. Because

$\tilde A$8

the upper bound comes from covering $\tilde A$9 by the $A$0 sets $A$1 at level $A$2, while the lower bound is obtained by pushing forward the Moran probability measure and applying the mass-distribution principle. Balls of radius $A$3 meet only finitely many $A$4 of comparable size, giving a bound of the form $A$5.

Yu then formulates the Fractal Quasi-Equivalence Principle explicitly: small perturbations of the defining similitudes, measured by $A$6, induce only small changes in the invariant set in the shape metric and preserve its geometry. In particular, for every $A$7 there is a one-to-one correspondence

$A$8

with

$A$9

and

$\hat h$0

Conversely, any two self-similar sets with small shape-difference are generated by IFSSs whose ratios and translations differ by at most $\hat h$1 (Yu, 2009).

Yu’s examples illustrate both robustness and limits. Perturbations of the middle-third Cantor system obtained by replacing the maps $\hat h$2 and $\hat h$3 by

$\hat h$4

produce a $\hat h$5-perturbation with $\hat h$6; all such perturbations still have Hausdorff dimension $\hat h$7 and positive finite $\hat h$8-measure. Analogous statements hold for rotated planar Cantor constructions, and for von Koch and Sierpiński-type examples with small shear, rotation, or twist at each scale. The principle fails when the perturbation is too large: if $\hat h$9, a Cantor set can be perturbed into an ordinary line segment, and the fractal dimension may be lost entirely (Yu, 2009).

A common misconception is that quasi-equivalence means exact geometric rigidity. Yu’s counterexample regime shows the opposite: the principle is perturbative and genuinely thresholded, with $\mathbb{R}^n$0 essential.

4. Operator-theoretic analogue: quasi-unitary equivalence of energy forms

A distinct but structurally related notion appears in the approximation of fractals by graph-like spaces. Post and Simmer define $\mathbb{R}^n$1-quasi-unitary equivalence for closed non-negative quadratic forms $\mathbb{R}^n$2 and $\mathbb{R}^n$3 with form domains $\mathbb{R}^n$4 and resolvents $\mathbb{R}^n$5. Bounded operators

$\mathbb{R}^n$6

and their restrictions $\mathbb{R}^n$7 exhibit $\mathbb{R}^n$8-quasi-unitary equivalence if $\mathbb{R}^n$9 is almost isometric, $\varphi,\psi$0 is almost its adjoint, $\varphi,\psi$1 and $\varphi,\psi$2 are almost identities in the form-domain norms, $\varphi,\psi$3 and $\varphi,\psi$4 differ by at most $\varphi,\psi$5 in the form norms, and the forms almost commute through $\varphi,\psi$6 and $\varphi,\psi$7. Equivalently,

$\varphi,\psi$8

for some explicit $\varphi,\psi$9. If $\hat h_{\rm rigid}$0 and all spaces coincide, this is exact unitary equivalence; if moreover $\hat h_{\rm rigid}$1, it reduces to $\hat h_{\rm rigid}$2 (Post et al., 2018).

The relation is transitive: if $\hat h_{\rm rigid}$3 and $\hat h_{\rm rigid}$4, then $\hat h_{\rm rigid}$5 with

$\hat h_{\rm rigid}$6

A corollary gives norm-resolvent control,

$\hat h_{\rm rigid}$7

In the principal application, a symmetric post-critically finite fractal $\hat h_{\rm rigid}$8 with self-similar Dirichlet form $\hat h_{\rm rigid}$9 is approximated first by finite weighted graphs $A,B\subset X$00, then by metric graphs $A,B\subset X$01, and finally by graph-like manifolds $A,B\subset X$02. There is an explicit

$A,B\subset X$03

such that the discrete graph form $A,B\subset X$04 is $A,B\subset X$05-quasi-unitarily equivalent to $A,B\subset X$06. Under compatibility conditions

$A,B\subset X$07

the rescaled metric-graph energy $A,B\subset X$08 is $A,B\subset X$09-quasi-unitarily equivalent to $A,B\subset X$10. For graph-like manifolds, under uniform compatibility and shrinking cross-section,

$A,B\subset X$11

By transitivity, both $A,B\subset X$12 and $A,B\subset X$13 converge to $A,B\subset X$14 (Post et al., 2018).

The consequences are spectral as well as geometric. Norm-resolvent control yields convergence of heat semigroups in operator norm, convergence of spectral projectors for spectral windows disjoint from $A,B\subset X$15, Hausdorff convergence of spectra, and convergence of eigenvalues and eigenfunctions in energy norm and in $A,B\subset X$16. This extends the language of quasi-equivalence from attractor geometry to Laplacians and energy forms.

5. Symbolic quasi-equivalence in Perron, Engel, and Pierce expansions

Moroz develops another formulation in the setting of number expansions. For a rule $A,B\subset X$17, the positive and alternating Perron expansions define cylinders

$A,B\subset X$18

and both cylinder types have the same length

$A,B\subset X$19

Using refined covering families $A,B\subset X$20 and $A,B\subset X$21 consisting of finite unions of consecutive cylinders of a given rank, Moroz proves faithfulness: $A,B\subset X$22 for all relevant subsets $A,B\subset X$23 (Moroz, 3 Oct 2025).

The fractal equivalence principle for Perron expansions is exact. The bijection

$A,B\subset X$24

is measure-preserving on cylinders and hence preserves Hausdorff dimension: $A,B\subset X$25

The fractal quasi-equivalence principle arises when the expansions differ. For classical versus modified Engel expansions, the natural map is the digit shift

$A,B\subset X$26

This map is Lipschitz but not bi-Lipschitz on all of $A,B\subset X$27, so in general only

$A,B\subset X$28

holds. Under the growth hypothesis

$A,B\subset X$29

and on the restricted set

$A,B\subset X$30

one recovers equality: $A,B\subset X$31

For Pierce expansions in Perron and traditional notation, the shift is

$A,B\subset X$32

and the relevant growth condition is weaker: $A,B\subset X$33 If

$A,B\subset X$34

then for every $A,B\subset X$35 with the extra condition “all $A,B\subset X$36,”

$A,B\subset X$37

The proofs combine faithful coverings with cylinder-length estimates. The upper bound comes from Lipschitz control on cylinders, while the lower bound uses the fact that on the restricted sets the ratio $A,B\subset X$38 or its Pierce analogue stays uniformly away from $A,B\subset X$39 at sufficiently deep ranks (Moroz, 3 Oct 2025).

These results clarify that quasi-equivalence need not mean a global metric equivalence of codings. Dimension preservation may survive even when the underlying map fails to be bi-Lipschitz, provided the admissible digit growth compensates for the symbolic distortion.

6. Fractal transference in continued fractions

Nakajima and Takahasi formulate a related principle for continued fractions. Let

$A,B\subset X$40

so that $A,B\subset X$41 is an injection $A,B\subset X$42. Good and Ramharter showed that $A,B\subset X$43. The main transference theorem states that if $A,B\subset X$44 has positive upper density, then there exists $A,B\subset X$45 with

$A,B\subset X$46

such that for every $A,B\subset X$47,

$A,B\subset X$48

If $A,B\subset X$49, there is similarly $A,B\subset X$50 with $A,B\subset X$51 and

$A,B\subset X$52

Thus density-combinatorics properties of subsets of $A,B\subset X$53 transfer to digit-value patterns of continued fractions on a set of Hausdorff dimension $A,B\subset X$54 (Nakajima et al., 15 Feb 2025).

A relative version applies to sets of polynomial density exponent $A,B\subset X$55. If

$A,B\subset X$56

for large $A,B\subset X$57, then for every $A,B\subset X$58 with positive relative upper density there is a compact set

$A,B\subset X$59

with

$A,B\subset X$60

such that

$A,B\subset X$61

for all $A,B\subset X$62.

The proof uses Moran-type seed sets $A,B\subset X$63, a sparse insertion scheme, and an elimination map

$A,B\subset X$64

that deletes the inserted digits. The critical step is dimension preservation: the elimination map is shown to be “almost Lipschitz,” meaning Hölder of exponent $A,B\subset X$65 on sufficiently deep Cantor levels for every $A,B\subset X$66. Combined with standard Hölder-dimension estimates, this prevents the insertion process from lowering the dimension below that of the seed set (Nakajima et al., 15 Feb 2025).

The transference theorem recovers arithmetic progressions from Szemerédi’s theorem, polynomial progressions from Bergelson–Leibman, and relative analogues over the primes, primes of the form $A,B\subset X$67, and Piatetski–Shapiro sequences. It also has a declared limitation: the monotonic-digit subspace

$A,B\subset X$68

has the opposite behavior, and positive-density constraints on $A,B\subset X$69 force a strict dimension drop. This makes the non-monotonicity and large-gap structure of the injective-digit set essential (Nakajima et al., 15 Feb 2025).

7. Limits, misconceptions, and open directions

Across these settings, quasi-equivalence is a stability principle, not an identity principle. In Yu’s geometric theory, the threshold $A,B\subset X$70 is essential; beyond it, a self-similar fractal can degenerate into a non-fractal object such as a line segment (Yu, 2009). In the operator-theoretic theory, $A,B\subset X$71-quasi-unitary equivalence is weaker than exact unitary equivalence and is designed to control resolvents, semigroups, spectral projectors, and eigenfunctions rather than to identify spaces literally (Post et al., 2018). In the symbolic theory of expansions, the relevant maps may be Lipschitz but not bi-Lipschitz globally, so equality of Hausdorff dimension may require explicit growth hypotheses on the digits (Moroz, 3 Oct 2025).

Yu also proposes a broader analogy with non-crystalline solids and regular patterns. The questions include: given a diffraction-regular crystal model, how large a $A,B\subset X$72-perturbation still produces an essentially discrete diffraction pattern; what is the supremal $A,B\subset X$73 beyond which long-range order is lost, called the collapse value; whether different substances admit the same collapse threshold; how elastic, electric, and optical properties change quantitatively with $A,B\subset X$74; whether there exist bodies in $A,B\subset X$75 whose maximal packing density jumps discontinuously under arbitrarily small shape perturbations, the collapse property; and, in tilings and patterns, when $A,B\subset X$76-quasi-prototiles remain quasi-isohedral or when tile-transitivity is lost (Yu, 2009).

Taken together, these formulations show that the fractal quasi-equivalence principle has evolved into a unifying idiom for perturbative invariance. In Euclidean self-similarity it concerns attractors and Hausdorff measure; in analysis on fractals it concerns Dirichlet forms and spectral convergence; in number expansions it concerns faithful coverings and digit-shift maps; and in continued fractions it becomes a transference mechanism from density combinatorics to Hausdorff dimension. The common thread is precise control of distortion sufficient to preserve a chosen fractal invariant, together with explicit conditions under which that control breaks down.