Moran-type Attractors Overview

• Moran-type attractors are invariant compact sets generated by non-stationary iterated function systems with varying contraction ratios and branching numbers.
• Their construction leverages advanced separation regimes and multifractal spectra to extend classical self-similar and Cantor set models.
• Dimension theory and spectral criteria of these attractors provide actionable insights into box-counting, Hausdorff, and Assouad dimensions in non-homogeneous settings.

A Moran-type attractor is a compact set generated as the invariant set of a non-stationary (often infinite) sequence of iterated function systems, where the contraction ratios and the branching numbers may vary from stage to stage. This generalizes the classical self-similar (or self-affine) attractors of autonomous IFSs to a much broader context encompassing non-homogeneous and non-stationary geometric constructions. Moran-type attractors and their invariant measures unify many classical and modern examples of fractal sets, such as non-homogeneous Cantor sets and rapidly inhomogeneous self-similar constructions. Their dimension theory, measure-theoretic properties, and spectrality theory have been developed to include generalized contraction maps, separation regimes, and multifractal spectra.

1. Definition of Moran-type Attractors

Moran-type attractors are associated with Moran-type iterated function systems (MIFS), which are defined as follows. Let $(X, \rho)$ denote a compact metric space. For each $n \geq 1$, let $\Phi_n = \{\phi_{n,1}, \dots, \phi_{n,N_n}\}$, with $2 \leq N_n < \infty$, be a finite family of bi-Lipschitz maps $\phi_{n,j}: X \to X$. Require constants $0 < c_{1,n} \leq c_{2,n} < 1$ such that for all $x, y \in X$,

$$c_{1,n}\, \rho(x, y) \leq \rho(\phi_{n,j}(x), \phi_{n,j}(y)) \leq c_{2,n} \, \rho(x, y),$$

and

$\lim_{n \to \infty} \prod_{i=1}^n c_{2,i} = 0.$

The family $\{\Phi_n\}_{n \geq 1}$ forms a MIFS. The Moran-type attractor (or invariant set) at level $n$ is the unique compact set $K_n \subset X$ such that

$K_n = \bigcup_{j=1}^{N_n} \phi_{n, j}(K_{n+1}),$

with

$$K_n = \left\{ \lim_{k \to \infty} \phi_{n, J_k}(a): J_k \in \Sigma_n^k,\, a \in X \right\}, \quad \text{where } \phi_{n, J} := \phi_{n, j_n} \circ \cdots \circ \phi_{n+k-1, j_{n+k-1}}$$

and $$\Sigma_n^k = \prod_{i=0}^{k-1} \{1, \dots, N_{n+i}\}$$ is the symbolic coding at stage $n$. In particular, the attractor $K_1$ is called the (primary) Moran-type attractor of the system (Cao et al., 16 Jan 2026).

2. Construction of Invariant Measures

Corresponding to each attractor is an associated sequence of Moran-type measures. For each $n$, select a probability vector $\mathbf{p}_n = (p_{n,1}, \dots, p_{n,N_n})$ with $p_{n,j} > 0$ and $\sum_j p_{n,j} = 1$. The product measure $\nu_n$ on the infinite code space $\Sigma_n^\mathbb{N}$ induces a Borel probability measure $\mu_n$ on $K_n$ via the canonical coding map $\pi_n$. The measure $\mu_n$ is the unique solution to

$$\mu_n = \sum_{j=1}^{N_n} p_{n,j}\, (\mu_{n+1} \circ \phi_{n,j}^{-1}),$$

with $\operatorname{supp} \mu_n = K_n$. The entire measure-theoretic structure is recursively determined by the measures at each stage, providing a measure-theoretic refinement of the attractor's geometric construction (Cao et al., 16 Jan 2026).

3. Separation Regimes and Regularity Properties

The dimension theory and structure of Moran-type attractors critically depend on separation properties:

• Moran-type Open-Set Condition (MOSC): There exist open sets $V_n \subset X$ such that images $\{\phi_{n,j}(V_{n+1})\}_j$ are contained in $V_n$ and have pairwise disjoint interiors, with $\inf_n \operatorname{Leb}^d(V_n) > 0$.
• Weak Separation Condition (MWSC): For suitable sets $U_n \subset X$, the number of overlapping images of equivalently contracted maps remains uniformly bounded at each stage.
• Strong Separation Condition (MSSC): At every level, the cylinder sets $\phi_{1, J}(K_{n+1})$ are pairwise disjoint.

The validity of some separation regime is often assumed in the proofs of exact dimension formulae and the identification of multifractal properties (Cao et al., 16 Jan 2026).

4. Dimension Theory of Moran-type Attractors

For Moran-type attractors generated by similarities $\phi_{n, j}$ of contraction ratios $r_{n, j}$, several dimension formulae are established:

• Box-Counting and Packing Dimensions: When the MWSC and a uniform lower contraction bound hold,

$$\underline{\dim}_B K_1 = \liminf_{b \to 0} \frac{\ln \# \mathcal{A}_b}{-\ln b}, \qquad \overline{\dim}_B K_1 = \dim_P K_1 = \limsup_{b \to 0} \frac{\ln \# \mathcal{A}_b}{-\ln b}$$

where $\mathcal{A}_b$ is the set of all finite compositions whose contraction ratio is at most $b$.

• Hausdorff Dimension via Pressure: Define $P_n(s) = \sum_{j=1}^{N_n} r_{n, j}^s$, $S_n(s) = \prod_{i=1}^n P_i(s)$. Under MOSC, if

$$\liminf_{n \to \infty} S_n(s) \in (0, \infty), \quad \text{then} \quad \dim_H K_1 = s.$$

Otherwise, $\dim_H K_1$ can be estimated via the asymptotics of $S_n(s)$ (Cao et al., 16 Jan 2026).

• Specific Formula for Staggered IFS: For the so-called “staggered contraction ratio” systems with $n_j = 2p_j$ and contraction bases $b_k > 2p_k > 2$,

$$\dim_H(A) = \liminf_{k \to \infty} \frac{\sum_{j=1}^k \ln(2p_j)}{\sum_{j=1}^k \ln b_j}.$$

This formula generalizes the classical Moran equation and applies even in highly inhomogeneous settings (Luo et al., 2024).

5. Spectrality and Structural Decomposition

A central question concerns when the invariant measure of a Moran-type attractor is spectral, i.e., admits an orthonormal basis of complex exponentials in $L^2(\mu)$.

• Spectrality Criterion: For the IFS

$$\phi_{k,d}(x) = (-1)^d b_k^{-1} (x + d), \quad d \in D_{2p_k} = \{0, \dots, 2p_k-1\}$$

if all $n_k = 2p_k$ are even, $\{b_k\}$ is bounded, and $2p_k \mid b_k$ for all $k \geq 2$, then the corresponding measure $\mu$ is spectral (Luo et al., 2024). The spectrum can be described as

$$\Lambda = D_{2p_1} + b_1 D_{2p_2} + b_1 b_2 D_{2p_3} + \cdots.$$

This generalizes the spectrality results for homogeneous Cantor measures.

• Fourier and Convolutional Structure: The Fourier transform of $\mu$ admits the infinite product representation

$$\widehat\mu(t) = \exp(-2\pi i\,b_1^{-1}t)\,\prod_{k=1}^\infty f_k \Bigl( \frac{t}{b_1\cdots b_k} \Bigr ),$$

where $$f_k(u) = \frac{1}{2p_k} \sum_{d=0}^{2p_k-1} (-1)^d e^{2\pi i d u}$$. This analytic structure underpins the proofs of spectrality (Luo et al., 2024).

• Hadamard Triple and Random Convolution Arguments: The proof of spectrality leverages specific “Hadamard triple” constructions at each level, facilitating a stepwise buildup of the exponential basis via convolutional factorization (Luo et al., 2024).

6. Assouad-type Dimensions and Multifractal Properties

Recent results have extended the dimension theory of Moran-type attractors to the Assouad dimension, lower dimension, and their corresponding spectra:

• Assouad Dimension for a homogeneous Moran set $E$ generated by parameters $\{n_k\}$ and $\{c_k\}$ with $\sup n_k < \infty$,

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

(Li et al., 2024). The lower dimension is bounded above by the corresponding $\liminf$.

• Spectra for Cantor-like Sets: For a more general class including inhomogeneous gaps, the Assouad and lower spectra admit explicit sup/inf characterizations in terms of scaling exponents over variable scales (Li et al., 2024).
• Connection to Doubling Measures and Separation: The validity of these dimension formulae requires finite branching, uniform contraction, and sufficient separation (typically, disjoint interiors at each level) (Li et al., 2024).

7. Illustrative Examples and Special Cases

• Non-homogeneous Cantor-type Constructions: Moran-type attractors include classically self-similar sets as special cases but allow for rapidly changing contractions and branching. For example, systems with $\phi_{n,0}(x) = \tfrac{1}{2}x$, $\phi_{n,1}(x) = \frac{1}{2}(x + \frac{1}{n}\rho^n)$ generate attractors with identical box and Hausdorff dimensions yet can have vanishing $\mathcal{H}^s$-measure at the dimension $s$ (Cao et al., 16 Jan 2026).
• Staggered IFS and Spectral Measures: For the class $\phi_{k,d}(x) = (-1)^d b_k^{-1}(x + d)$ with even $n_k$ and divisibility $2p_k \mid b_k$, the resulting Moran-type measure exhibits spectrality, contrasting with general inhomogeneous self-similar measures, which typically fail to be spectral unless strong arithmetic conditions are imposed (Luo et al., 2024).
• Assouad Spectrum Computations: Explicit formulae for Assouad and lower spectra are established for homogeneous and Cantor-like sets under finite-branching and separation, extending and refining the classical results for stationary self-similar sets (Li et al., 2024).

A plausible implication is that Moran-type attractors provide a flexible modeling framework for both geometric and spectral phenomena beyond the homogeneous, self-similar setting, enabling systematic investigation of the interplay between geometric inhomogeneity, measure regularity, and harmonic analysis.