Moran-type Self-Similar Sets

• Moran-type self-similar sets are fractal limit sets defined via recursive tree constructions that subdivide a compact set into non-overlapping similar copies with variable contraction ratios.
• Key developments include formulating separation conditions like OSC, SSC, and WSC alongside the classical Moran equation to determine the similarity and fractal dimensions.
• Advanced studies encompass diverse dimension theories—including Hausdorff, box-counting, Assouad, and intermediate dimensions—as well as generalizations to non-autonomous, affine, and random settings.

A Moran-type self-similar set is a limit set generated by a recursive tree-like construction, where at each level $k\geq1$ a compact set $J$ with nonempty interior is subdivided into $n_k$ non-overlapping similar copies, called "basic sets," each scaled by specified contraction ratios $(c_{k,1},\ldots,c_{k,n_k})$. This framework generalizes classical autonomous self-similar sets by allowing both the number of pieces and contraction ratios to vary at every generative step, and can further incorporate digit or block restrictions in symbolic representations. The theory of Moran sets encompasses their combinatorial structure, geometric and fractal properties, and a rich multi-parameter dimension theory beyond the Hausdorff dimension—including packing, box-counting, Assouad, and intermediate dimensions. Numerous extensions exist to random, affine, perturbative, and measure-theoretic settings.

1. Defining Moran-type Self-Similar Sets

Given a compact set $J\subset\mathbb{R}^d$ with $|J|=1$ (diameter), integer sequences $\{n_k\}$ ($n_k\geq2$) and contraction ratios $\Omega_k=(c_{k,1},\ldots,c_{k,n_k})$ with $0<c_{k,i}<1$, one defines the index sets

$$D_0=\{\emptyset\},\qquad D_k=\{u_1u_2\cdots u_k\,:\,1\leq u_j\leq n_j\},\qquad D=\bigcup_{k\geq0}D_k,$$

and constructs a tree $\{J_u:u\in D\}$ under the Moran structure conditions (MSC):

• For each $u\in D$, there exists a similarity $S_u:\mathbb{R}^d\to\mathbb{R}^d$ such that $J_u=S_u(J)$, starting with $J_{\emptyset}=J$.
• For $u\in D_{k-1}$, the children $\{J_{u\,i}:1\leq i\leq n_k\}$ inside $J_u$ have pairwise disjoint interiors, and $|J_{u\,i}|/|J_u|=c_{k,i}$.

The resulting Moran set is

$E=\bigcap_{k=1}^{\infty}\;\bigcup_{u\in D_k}J_u,$

sometimes denoted $\mathcal{M}(J,\{n_k\},\{\Omega_k\})$. The homogeneous case has $c_{k,i}$ independent of $i$ at each level; inhomogeneity or further restrictions induce Cantor-like or block-restricted sets (Li et al., 2014, Serbenyuk, 2017).

2. Separation Conditions and the Moran Equation

Fundamental geometric and measure-theoretic properties depend on separation conditions imposed upon the construction:

• Open Set Condition (OSC): at each $k$, there exists a non-empty open $U$ with $S_{u\,i}(U)\subset U$ and $S_{u\,i}(U)\cap S_{u\,j}(U)=\emptyset$ for $i\neq j$.
• Strong Separation Condition (SSC): images $S_{u\,i}(E)\cap S_{u\,j}(E)=\emptyset$ at each level.
• Weak Separation Condition (WSC): intrinsic non-overlap in the interiors of basic sets at each generation, encoded in the "level separation property" (Sánchez-Granero et al., 2017).

The classic Moran equation describes the similarity dimension $s$: $\sum_{j=1}^{n_k}\,c_{k,j}^s\,=\,1,$ with uniqueness for $c_{k,j}<1$ and $n_k\,c_{k,j}<1$ (Serbenyuk, 2017, Rajala et al., 2010). Under OSC or suitable variants, Hausdorff, box, and packing dimensions equal the solution $s$ (Cao et al., 16 Jan 2026).

3. Dimension Theory: Hausdorff, Packing, Box, Assouad

For a set $E$ defined as above, multiple dimension notions arise:

• Hausdorff dimension: Under OSC and bounded contraction ratios, $\dim_H\,E$ is the unique $s$ solving the Moran equation. In general,

$$\frac{\log\,n_{k+1} \cdots n_{k+m}}{-\log\,c_{k+1} \cdots c_{k+m}}$$

arises as an explicit formula in stationary and inhomogeneous cases (Li et al., 2014, Miao et al., 12 Nov 2025).

• Box and Packing dimension: For suitable weak separation, lower and upper box dimensions $\underline{\dim}_B\,E$, $\overline{\dim}_B\,E$ and packing dimension $\dim_P\,E$ are given by liminf and limsup versions of the above ratio (Gu et al., 2023, Cao et al., 16 Jan 2026).
• Assouad dimension: Measures worst-case local scaling. For $\inf_{k,i}c_{k,i}>0$,

$$\dim_A\,E = \lim_{m \to \infty} \sup_{k \geq 1} \frac{\log(n_{k+1} \cdots n_{k+m})}{-\log(c_{k+1,1} \cdots c_{k+m,n_{k+m}})},$$

with analogous formulas for Cantor-like sets with ratio-perturbations, and explicit lower bounds for quasi-Assouad (see (Li et al., 2014, Miao et al., 12 Nov 2025)).

A table of canonical dimension formulas:

Dimension	Formula (homogeneous)	Key Condition
Hausdorff	$$\lim_{m\to\infty}\sup_k \frac{\log(n_{k+1}\cdots n_{k+m})}{-\log(c_{k+1}\cdots c_{k+m})}$$	OSC, $c_*>0$
Assouad	$$\lim_{m\to\infty} \sup_k \frac{\log(n_{k+1}\cdots n_{k+m})}{-\log(c_{k+1}\cdots c_{k+m})}$$	$c_*>0$
Packing/Box	$$\liminf/\limsup_{n \to \infty} \frac{\sum_{i=1}^n \log n_i}{-\sum_{i=1}^n \log r_i}$$	MWSC/MOSC

In many cases (OSC + $c_*>0$), all these dimensions coincide. Otherwise, strict inequalities can arise between Hausdorff, packing, and Assouad dimensions (Li et al., 2014, Miao et al., 12 Nov 2025, Gu et al., 2023).

4. Generalizations: Non-Autonomous, Controlled, Affine and Random Moran Constructions

• Non-autonomous IFS: Contraction mappings may vary at every step, forming non-stationary or time-dependent Moran sets. The limit set theory and dimension formulas generalize using topological pressure and Bowen's formula (Rempe-Gillen et al., 2012).
• Controlled/Weakly Controlled Moran Constructions (CMC/WCMC): Eschewing exact similarity mapping, one adopts multi-level diameter constraints and separation via finite clustering or ball conditions, allowing broader constructions including self-affine and sub-self-affine sets (Rajala et al., 2010, Käenmäki et al., 2017). In doubling metric spaces, finite clustering, ball condition, and open set condition are equivalent for the positivity of critical-dimensional Hausdorff measure.
• Affine and self-affine Moran sets: At each step, more general affine transformations (including different scaling in each coordinate) yield limits whose dimensions must be estimated via combinatorial covering and entropy arguments (Gu et al., 2023).
• Random Moran constructions: If contraction ratios and/or weights are randomly selected (i.i.d. at each step), almost sure formulas for Assouad-like or $\Phi$-dimensions exist, sometimes exhibiting dimension gaps between measures and sets due to independence or dependence of weights and ratios (Hare et al., 26 Mar 2025).

5. Intermediate, Quasi-Assouad, and Other Fractal Spectra

Recent work extends dimension theory beyond classical notions:

• Intermediate dimensions: For $\theta\in[0,1]$, these interpolate between Hausdorff ($\theta=0$) and box-counting ($\theta=1$) dimensions. In Moran sets with $c_*>0$, upper and lower intermediate dimensions are computed via cut-set sum formulas depending on the scales $|J_u|$ and the parameter $\theta$ (Du et al., 2024).
• Quasi-Assouad dimensions: These weaken Assouad dimension by permitting mild scale irregularities at each level (controlled by $\eta>0$), with exact formulas available under bounded branching, quasi-normality, or thin-fat control (Miao et al., 12 Nov 2025).
• New fractal-structure dimensions: Finite-covering definitions (e.g., $\dim_\Gamma^3$, $\dim_\Gamma^4$) often coincide with similarity dimension under new "weak separation" conditions, sometimes strictly weaker than OSC (Sánchez-Granero et al., 2017).

Visualization and Möbius parametrizations demonstrate rich spectra, especially in block-periodic or non-homogeneous constructions, where the upper intermediate dimension is a Möbius function of $\theta$ (Du et al., 2024).

6. Measures, Topology, and Structure Theory

• Self-similar Moran measures: Assigning probability weights $p_{k,i}$ at each step yields invariant measures with exact dimension formulas (entropy dimension for self-affine, exact in homogeneous settings) (Serbenyuk, 2018, Gu et al., 2023, Cao et al., 16 Jan 2026).
• Topological properties: Under OSC, Moran sets are perfect, nowhere dense, totally disconnected, and carry zero Lebesgue measure except in degenerate cases (Serbenyuk, 2017). Digit- and block-restricted variants are homeomorphic to Cantor sets.
• Symbolic and metric structure: The symbolic coding of paths through the hierarchy is essential, enabling identification of Moran sets as boundaries of Gromov-hyperbolic trees (Luo, 2012), with explicit homeomorphism and (under rearrangeability) Lipschitz equivalence. These trees encode combinatorial and geometric complexity underlying Moran sets and their extensions.

7. Perturbations, Homogeneity, and Further Developments

• Perturbed/quasi-self-similar Moran sets: Permitting per-step errors (e.g. $\delta$-perturbations of ratios or placements) yields sets with the same dimension as the unperturbed case, provided an open-set condition holds (Yu, 2009).
• Cantor-like and Furstenberg-homogeneous constructions: Cantor-like sets with small deviations in similarity ratios retain classical dimension properties, while Furstenberg homogeneity ties the weak separation condition to dynamical properties of self-similar sets in $\mathbb{R}$ (Käenmäki et al., 2015).

Open directions include classification of dimension spectra for non-autonomous and random settings, comparative rigidity under Lipschitz or quasi-symmetric equivalence, and dynamical or multifractal analysis of measures and associated stochastic processes.

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Key papers referenced: (Li et al., 2014, Serbenyuk, 2017, Rajala et al., 2010, Cao et al., 16 Jan 2026, Gu et al., 2023, Miao et al., 12 Nov 2025, Luo, 2012, Käenmäki et al., 2015, Sánchez-Granero et al., 2017, Serbenyuk, 2018, Du et al., 2024, Rempe-Gillen et al., 2012, Hare et al., 26 Mar 2025, Käenmäki et al., 2017, Yu, 2009, Jiang et al., 2021).