--- title: Fractal Quasi-Equivalence Principle url: https://www.emergentmind.com/topics/fractal-quasi-equivalence-principle type: topic --- # Fractal Quasi-Equivalence Principle 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 [0902.1615]. Related formulations appear in operator theory, where quasi-unitary equivalence compares energy forms on fractals, graphs, metric graphs, and graph-like manifolds [1802.02998], 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 [2510.03373], [2502.10902]. 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 \[ r(A)=\inf_{x\in\mathbb{R}^n}\sup_{a\in A}|x-a|, \qquad r(B)=\inf_{x\in\mathbb{R}^n}\sup_{b\in B}|x-b|, \] Yu normalizes by scale and defines the **relative shape difference** \[ \tilde h(A,B)=\hat h\!\Bigl(\tfrac1{r(A)}A,\tfrac1{r(B)}B\Bigr), \] 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** $\tilde h_{\rm rigid}$; it is again complete [0902.1615]. 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 $\delta$-quasi-self-similarity, quasi-prototiles, and quasi-motifs. ## 2. $\delta$-quasi-self-similar sets Fix an iterated-function-system of similitudes $\mathcal{S}=\{S_i\}_{i=1}^m$, where each $S_i:\mathbb{R}^n\to\mathbb{R}^n$ has Lipschitz ratio $c_i<1$. Its invariant self-similar set $E$ satisfies \[ E=\bigcup_{i=1}^m S_i(E). \] For a multi-index $i=(i_1\cdots i_k)$, Yu writes \[ S_i=S_{i_1}\circ\cdots\circ S_{i_k},\qquad c_i=c_{i_1}\cdots c_{i_k},\qquad E_i=S_i(E). \] A perturbation is described by a **structure system** $\{F_i\}$ for a compact set $F$, indexed by finite words, such that \[ F_i=\bigcup_{j=1}^m F_{ij},\qquad F_\varnothing=F. \] Given nonnegative errors $\{\delta_i\}$ with $\delta=\sup_i\delta_i<\infty$, Yu calls $F$ a **$\{\delta_i\}$-perturbation** of $E$, or **$\delta$-quasi-self-similar**, if \[ h(F_i,E_i)\le \delta_i\,c_i\,r(E) \quad\text{for every multi-index }i. \] When $\delta<1$, one speaks of a $\delta$-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 $\{V_i\}$ such that: (i) $F_i\subset\overline{V_i}$; (ii) $V_i\cap V_j=\emptyset$ for $i\ne j$; (iii) each $V_i$ contains a ball of radius $a_1c_i$ and lies in a ball of radius $a_2c_i$ [0902.1615]. This formulation is stricter than a vague “small perturbation” heuristic. The perturbation is controlled at every cylinder level, scaled by the contraction factor $c_i$, and measured against the radius $r(E)$ 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 $s$ be the unique solution of \[ \sum_{i=1}^m c_i^s=1. \] If $F$ is a $\delta$-quasi-self-similar set satisfying the Open Set Condition, then \[ \dim_H F=\dim_B F=s, \qquad 0<\mathcal{H}^s(F)<+\infty \] [0902.1615]. The proof follows the classical Moran scheme but with perturbed cylinder sets. Because \[ \operatorname{diam}(F_i)\le(1+\delta)\,\operatorname{diam}(E)\,c_i, \] the upper bound comes from covering $F$ by the $m^k$ sets $F_i$ at level $k$, while the lower bound is obtained by pushing forward the Moran probability measure and applying the mass-distribution principle. Balls of radius $r$ meet only finitely many $F_i$ of comparable size, giving a bound of the form $p(B(x,r))\lesssim r^s$. Yu then formulates the **Fractal Quasi-Equivalence Principle** explicitly: small perturbations of the defining similitudes, measured by $\delta$, induce only small changes in the invariant set in the shape metric and preserve its geometry. In particular, for every $\delta<1$ there is a one-to-one correspondence \[ E\longleftrightarrow F \] with \[ \hat h(E,F)\le \frac{\delta}{1-\max_i c_i}\,r(E), \] and \[ \dim_H F=\dim_H E,\qquad 0<\mathcal{H}^{\dim_H E}(F)<\infty. \] Conversely, any two self-similar sets with small shape-difference are generated by IFSSs whose ratios and translations differ by at most $O(\hat h)$ [0902.1615]. Yu’s examples illustrate both robustness and limits. Perturbations of the middle-third Cantor system obtained by replacing the maps $x\mapsto x/3$ and $x\mapsto x/3+2/3$ by \[ x\mapsto \frac{x}{3}+a_i,\qquad x\mapsto \frac{x}{3}+b_i, \quad a_i\in[0,a_0],\;b_i\in[b_0,1], \] produce a $\delta$-perturbation with $\delta=O(\max\{|a_i|,|b_i-1|\})$; all such perturbations still have Hausdorff dimension $\log2/\log3$ and positive finite $\mathcal{H}^{\log2/\log3}$-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 $\delta\ge1$, a Cantor set can be perturbed into an ordinary line segment, and the fractal dimension may be lost entirely [0902.1615]. 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 $\delta<1$ 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 $\delta$-**quasi-unitary equivalence** for closed non-negative quadratic forms $\bigl(\mathcal{E}_1,\mathcal{H}_1\bigr)$ and $\bigl(\mathcal{E}_2,\mathcal{H}_2\bigr)$ with form domains $\mathcal{H}_i^1$ and resolvents $R_i=(\Delta_i+1)^{-1}$. Bounded operators \[ J:\mathcal{H}_1\to\mathcal{H}_2,\qquad J':\mathcal{H}_2\to\mathcal{H}_1 \] and their restrictions $J^1,J^{\prime1}$ exhibit $\delta$-quasi-unitary equivalence if $J$ is almost isometric, $J'$ is almost its adjoint, $J'J$ and $JJ'$ are almost identities in the form-domain norms, $J^1$ and $J$ differ by at most $\delta$ in the form norms, and the forms almost commute through $J^1$ and $J^{\prime1}$. Equivalently, \[ \|R_2J-JR_1\|\le C\,\delta \] for some explicit $C\ge1$. If $\delta=0$ and all spaces coincide, this is exact unitary equivalence; if moreover $J=\mathrm{id}$, it reduces to $\|R_1-R_2\|\le\delta$ [1802.02998]. The relation is transitive: if $\mathcal{E}_1\sim_\delta\mathcal{E}_2$ and $\mathcal{E}_2\sim_{\tilde\delta}\mathcal{E}_3$, then $\mathcal{E}_1\sim_{\hat\delta}\mathcal{E}_3$ with \[ \hat\delta\le \delta+\tilde\delta+C\,\delta\,\tilde\delta. \] A corollary gives norm-resolvent control, \[ \bigl\|R_1-J^\ast R_2J\bigr\|\le C\,\delta. \] In the principal application, a symmetric post-critically finite fractal $K\subset\mathbb{R}^d$ with self-similar Dirichlet form $\mathcal{E}_K$ is approximated first by finite weighted graphs $(G_m,\mu_m,\gamma_m)$, then by metric graphs $M_m$, and finally by graph-like manifolds $X_m$. There is an explicit \[ \delta_m=O\bigl((r/N)^{m/2}\bigr)\to0 \] such that the discrete graph form $\mathcal{E}_{G_m}$ is $\delta_m$-quasi-unitarily equivalent to $\mathcal{E}_K$. Under compatibility conditions \[ \sum_{e\in E_v}\ell_{m,e}=2\,\mu_m(v)\,c_m^{-2}, \qquad \ell_{m,e}\,\gamma_{m,e}=c_m^2\,\tau_m, \] the rescaled metric-graph energy $\tau_m\mathcal{E}_{M_m}$ is $\delta'_m$-quasi-unitarily equivalent to $\mathcal{E}_{G_m}$. For graph-like manifolds, under uniform compatibility and shrinking cross-section, \[ \delta''_m=\max\Bigl\{O\bigl(\rho_m/\ell_0\bigr),\;O\bigl(\ell_\infty^{-2}\rho_m^{-1}\bigr)\Bigr\}\xrightarrow[m\to\infty]{}0. \] By transitivity, both $\mathcal{E}_{M_m}$ and $\mathcal{E}_{X_m}$ converge to $\mathcal{E}_K$ [1802.02998]. 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 $\operatorname{spec}\Delta_K$, Hausdorff convergence of spectra, and convergence of eigenvalues and eigenfunctions in energy norm and in $L^2$. 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 $P=\{\varphi_n\}_{n\ge0}$, the positive and alternating Perron expansions define cylinders \[ \Delta^P_{c_1\cdots c_k} =\{x:p_1(x)=c_1,\dots,p_k(x)=c_k\}, \qquad \Delta^{P^-}_{c_1\cdots c_k} =\{x:q_1(x)=c_1,\dots,q_k(x)=c_k\}, \] and both cylinder types have the same length \[ \bigl|\Delta^P_{c_1\cdots c_k}\bigr| =\bigl|\Delta^{P^-}_{c_1\cdots c_k}\bigr| =\frac{r_0\,r_1\cdots r_{k-1}} {(c_1-1)c_1\,(c_2-1)c_2\cdots(c_k-1)c_k}. \] Using refined covering families $\mathfrak P$ and $\mathfrak P^-$ consisting of finite unions of consecutive cylinders of a given rank, Moroz proves faithfulness: \[ \dim_H(E,\mathfrak P)=\dim_H(E),\qquad \dim_H(E,\mathfrak P^-)=\dim_H(E) \] for all relevant subsets $E$ [2510.03373]. The **fractal equivalence principle** for Perron expansions is exact. The bijection \[ \mathcal F_P:\Delta^P_{c_1c_2\cdots}\longmapsto \Delta^{P^-}_{c_1c_2\cdots} \] is measure-preserving on cylinders and hence preserves Hausdorff dimension: \[ \dim_H\bigl(\mathcal F_P(E)\bigr)=\dim_H(E),\qquad E\subset(0,1]. \] The **fractal quasi-equivalence principle** arises when the expansions differ. For classical versus modified Engel expansions, the natural map is the digit shift \[ \mathcal T:\Delta^E_{c_1c_2\cdots}\mapsto \Delta^{E_{\rm mod}}_{(c_1+0)\,(c_2+1)\,(c_3+2)\cdots}, \qquad p'_n(\mathcal T(x))=p_n(x)+n-1. \] This map is Lipschitz but not bi-Lipschitz on all of $(0,1]$, so in general only \[ \dim_H\bigl(\mathcal T(E)\bigr)\le \dim_H(E) \] holds. Under the growth hypothesis \[ \sum_{n=1}^\infty \frac{n}{\psi(n)}<\infty \] and on the restricted set \[ \mathfrak A_\psi =\{x\in(0,1]: p_n(x)\ge\psi(n)\text{ for all large }n\}, \] one recovers equality: \[ \dim_H\bigl(\mathcal T(E)\bigr)=\dim_H(E) \quad\text{for every }E\subset\mathfrak A_\psi. \] For Pierce expansions in Perron and traditional notation, the shift is \[ \mathcal G:\Delta^{\rm Pierce}_{c_1c_2\cdots}\mapsto \Delta^{\rm Pierce}_{(c_1+1)\,(c_2+1)\cdots}, \qquad \tilde q_n(\mathcal G(x))=q_n(x), \] and the relevant growth condition is weaker: \[ \sum_{n=1}^\infty \frac{1}{\psi(n)}<\infty. \] If \[ \mathfrak B_\psi =\{x\in(0,1)\setminus\mathbb{Q}: q_n(x)\ge\psi(n)\text{ for all large }n\}, \] then for every $D\subset\mathfrak B_\psi$ with the extra condition “all $\tilde q_1\ge2$,” \[ \dim_H\bigl(\mathcal G(D)\bigr)=\dim_H(D). \] 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 $|\mathcal T(C)|/|C|$ or its Pierce analogue stays uniformly away from $0$ at sufficiently deep ranks [2510.03373]. 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 \[ E=\{x\in(0,1)\setminus\mathbb{Q}: a_m(x)\ne a_n(x)\text{ whenever }m\ne n\}, \] so that $n\mapsto a_n(x)$ is an injection $\mathbb{N}\to\mathbb{N}$. Good and Ramharter showed that $\dim_H(E)=1/2$. The main transference theorem states that if $S\subset\mathbb{N}$ has positive upper density, then there exists $E_S\subset E$ with \[ \dim_H(E_S)=\dim_H(E)=1/2 \] such that for every $x\in E_S$, \[ \overline\delta\bigl(\{a_n(x):n\in\mathbb{N}\}\cap S\bigr)=\overline\delta(S). \] If $d_B(S)>0$, there is similarly $F_S\subset E$ with $\dim_H(F_S)=1/2$ and \[ d_B\bigl(\{a_n(x):n\in\mathbb{N}\}\cap S\bigr)=d_B(S). \] Thus density-combinatorics properties of subsets of $\mathbb{N}$ transfer to digit-value patterns of continued fractions on a set of Hausdorff dimension $1/2$ [2502.10902]. A relative version applies to sets of polynomial density exponent $\alpha\ge1$. If \[ C_1\,N^{1/\alpha}/(\log N)^\beta \le |S\cap[1,N]| \le C_2\,N^{1/\alpha}/(\log N)^\beta \] for large $N$, then for every $A\subset S$ with positive relative upper density there is a compact set \[ E_{S,A}\subset\{x\in E:\{a_n(x)\}\subset S\} \] with \[ \dim_H(E_{S,A})=\frac1{2\alpha} \] such that \[ \overline\delta\bigl(\{a_n(x):n\in\mathbb{N}\}\cap A\,|\,S\bigr)=d(A|S) \] for all $x\in E_{S,A}$. The proof uses Moran-type seed sets $R(t,L;S^\ast)$, a sparse insertion scheme, and an elimination map \[ f:\text{target Cantor-like subset of }E \to R(t,L;S^\ast) \] that deletes the inserted digits. The critical step is dimension preservation: the elimination map is shown to be “almost Lipschitz,” meaning Hölder of exponent $1-\varepsilon$ on sufficiently deep Cantor levels for every $\varepsilon>0$. Combined with standard Hölder-dimension estimates, this prevents the insertion process from lowering the dimension below that of the seed set [2502.10902]. 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 $y^2+z^2+1$, and Piatetski–Shapiro sequences. It also has a declared limitation: the monotonic-digit subspace \[ J=\{x:a_1(x)