--- title: Homogeneous Moran Sets url: https://www.emergentmind.com/topics/homogeneous-moran-sets type: topic --- # Homogeneous Moran Sets Homogeneous Moran sets are the level-wise uniform subclass of Moran constructions: at each level \(k\), every parent has the same number of offspring, usually denoted \(N_k\) or \(n_k\), and all offspring at that level have the same contraction ratio \(c_k\), although both \(N_k\) and \(c_k\) may vary with \(k\) [1504.05354]. Across the literature, this class appears in Euclidean interval models, in overlap-allowing one-dimensional constructions, and in general complete doubling metric spaces; what remains common is the replacement of full self-similar stationarity by non-autonomous but level-uniform geometry [2409.06186][2407.14837][2005.06163]. Homogeneous Moran sets form a principal testing ground for explicit formulas for Hausdorff, packing, box, Assouad, quasi-Assouad, and intermediate dimensions, as well as for results on local dimensions of measures, arithmetic images, hyperbolic boundary models, and quasisymmetric rigidity [1504.05354][2511.09255][2510.00540]. ## 1. Definitions and model classes In the standard Euclidean formulation, one starts with a compact set \(J\subset \mathbb{R}^d\) with nonempty interior, symbolic address sets \(D_k\), and basic sets \(J_u\) that are geometrically similar to \(J\), nested inside their parents, and have pairwise disjoint interiors at each fixed level [2409.06186][1404.4409]. The homogeneous specialization is the condition \[ c_{k,i}=c_k \qquad \text{for every } i=1,2,\dots,n_k, \] so every level-\(k\) child has diameter \(c_k\) times the diameter of its parent [2409.06186]. In this setting the total number and common diameter of level-\(k\) basic sets are \[ N_k=\prod_{i=1}^k n_i, \qquad \delta_k=\prod_{i=1}^k c_i \] in the notation of one-dimensional interval models [2407.14837]. A more general metric-space formulation replaces Euclidean geometry by a complete doubling metric space \((X,d)\), uses a codetree \(\Sigma\), and defines a Moran construction by nested compact sets \(E_i\) with diameters tending to zero along branches, disjoint offspring, a uniform interior ball condition, and asymptotically negligible diameter fluctuations [1504.05354]. In that framework, a Moran construction is asymptotically homogeneous when it is asymptotically spatially symmetric and \[ c_{k,i}=c_k \qquad \text{for all } i\in\{1,\ldots,N_k\}. \] This is the exact analogue of the standard level-homogeneous Moran construction, but now valid in complete doubling metric spaces [1504.05354]. One-dimensional interval models often make the level data completely explicit. In the disjoint setting, a homogeneous Moran structure on \(I=[0,1]\) is a family of closed intervals \(I_\sigma\) such that each level-\((k-1)\) interval contains \(n_k\) children with pairwise disjoint interiors and \[ |I_{\sigma*i}|=c_k |I_\sigma| \qquad (1\le i\le n_k) \] [2407.14837]. In the overlap-allowing model \((\mathcal M,c_k,n_k,\kappa)\), the children still have common relative length \(c_k\), but overlaps are allowed with \[ 0\le \kappa<1, \] and the intersection of two level-\(k\) basic intervals is controlled relative to their length [2005.06163]. This distinction matters: some papers use “homogeneous Moran set” for the disjoint case, while others use it for a level-uniform construction with controlled overlaps. ## 2. Dimension formulas The central dimensional feature of homogeneous Moran sets is that the general pressure-like equation collapses to an explicit ratio of cumulative branching and cumulative contraction. In the asymptotically spatially symmetric metric-space theory, the finite-level quantity \(s_n\) is defined by \[ \prod_{k=1}^n\sum_{i=1}^{N_k} c_{k,i}^{s_n}=1. \] In the homogeneous case this becomes \[ \prod_{k=1}^n N_k c_k^{s_n}=1, \qquad s_n=\frac{\sum_{k=1}^n \log N_k}{-\sum_{k=1}^n \log c_k}, \] and the dimension formulas simplify to \[ \dim_H(E)=\liminf_{n\to\infty}\frac{\sum_{k=1}^n\log N_k}{-\sum_{k=1}^n\log c_k}, \] \[ \dim_P(E)=\dim_B(E)=\limsup_{n\to\infty}\frac{\sum_{k=1}^n\log N_k}{-\sum_{k=1}^n\log c_k}. \] In the stationary homogeneous case \(N_k\equiv N\), \(c_k\equiv c\), these reduce to \[ \dim_H(E)=\dim_P(E)=\dim_B(E)=\frac{\log N}{-\log c} \] [1504.05354]. The same reduction appears in broader thermodynamic and non-autonomous IFS frameworks. In the inhomogeneous Moran-set formalism based on pressure and pre-dimensions, the homogeneous case is characterized by \(\Psi_\omega^{(k)}=\Psi_{\omega'}^{(k)}\) for all \(\omega,\omega'\in D_k\), and the pressure-zero equation becomes \[ \#D_k \cdot (\Psi^{(k)})^{s_k}=1. \] If \(\Psi^{(k)}=\prod_{i=1}^k c_i\), then \[ s_k=-\frac{\sum_{i=1}^k \log n_i}{\sum_{i=1}^k \log c_i}, \] which is the same classical homogeneous Moran formula [1211.0927]. In the Moran-type IFS framework, the homogeneous specialization \(r_{n,j}=r_n\) likewise yields \[ \dim_H(K_1)=\liminf_{n\to\infty}\frac{\log(N_1\cdots N_n)}{-\log(r_1\cdots r_n)} \] [2601.11023]. A recent one-dimensional refinement with nonuniform boundary gaps gives a different but still explicit Hausdorff formula. For homogeneous Moran sets with levelwise constant left and right boundary gaps \(L_{k+1}\), \(R_{k+1}\), one defines \[ \delta_k=\prod_{i=1}^k c_i, \qquad e_{k+1}=\delta_k-L_{k+1}-R_{k+1}, \] and under one of three interior-gap conditions obtains \[ \dim_H E = \lim_{k\to\infty}\inf \frac{\log(n_1n_2\cdots n_k)}{-\log(\delta_k-L_{k+1}-R_{k+1})} \] [2510.00540]. This shows that once boundary trimming is built into the model, the effective scale may be \(e_{k+1}\) rather than \(\delta_k\). ## 3. Measures, local dimensions, and \(L^q\)-dimensions For measures on Moran constructions, local dimension can be read directly from nested cylinders. If \(\mu\) is supported on the limit set \(E\) of a Moran construction in a complete doubling metric space, then for \(\mu\)-almost all \(x_i\in E\), \[ \overline{\dim}_{\mathrm{loc}}(\mu,x_i) = \limsup_{n\to\infty} \frac{\log\mu(E_{i|_n})}{\log\operatorname{diam}(E_{i|_n})}, \] \[ \underline{\dim}_{\mathrm{loc}}(\mu,x_i) = \liminf_{n\to\infty} \frac{\log\mu(E_{i|_n})}{\log\operatorname{diam}(E_{i|_n})} \] [1504.05354]. The canonical measure in the homogeneous case is the uniformly distributed measure \[ \mu(E_i)=\prod_{k=1}^{n} N_k^{-1} \qquad \text{for all } i\in\Sigma_n. \] For an asymptotically homogeneous Moran construction, this measure has constant local and global \(L^q\)-dimensions: \[ \dim_q(\mu,x)=\dim_q(\mu) = \liminf_{n\to\infty} \frac{\sum_{k=1}^{n}\log N_k}{-\sum_{k=1}^{n}\log c_k} \qquad (q>1), \] \[ \dim_q(\mu,x)=\dim_q(\mu) = \limsup_{n\to\infty} \frac{\sum_{k=1}^{n}\log N_k}{-\sum_{k=1}^{n}\log c_k} \qquad (0\eta}} \frac{\log(n_{k+1}\cdots n_{k+l})}{-\log(c_{k+1}\cdots c_{k+l})} \] because all homogeneous Moran structures are quasi-normal [2511.09255]. If, in addition, \[ \liminf_{k\to\infty}\frac{\log c_k}{\log(c_1\cdots c_k)}>0 \] and BBC holds, then \[ \dim_{qA}E=\dim_AE \] [2511.09255]. Intermediate dimensions furnish a different interpolation between Hausdorff and box-counting behavior. For homogeneous Moran sets, define \(l(k,\theta)\) by \[ c_1c_2\cdots c_{l(k,\theta)} \le (c_1c_2\cdots c_k)^\theta < c_1c_2\cdots c_{l(k,\theta)-1}. \] Then, when \(c_*>0\), \[ \underline{\dim}_\theta E = \liminf_{k\to\infty} \min_{k\le m\le l(k,\theta)} \frac{\log(n_1\cdots n_m)}{-\log(c_1\cdots c_m)}, \] \[ \overline{\dim}_\theta E = \limsup_{k\to\infty} \min_{k\le m\le l(k,\theta)} \frac{\log(n_1\cdots n_m)}{-\log(c_1\cdots c_m)} \] [2409.06186]. These formulas show that even homogeneous Moran sets need not have a genuine intermediate dimension: one explicit example has \(c_k=\frac14\) and branching numbers alternating in huge factorial blocks between \(3\) and \(2\), with \[ \overline{\dim}_\theta E=\frac{\log 3}{2\log 2} \quad (\theta\in(0,1]), \qquad \underline{\dim}_\theta E=\frac12 \quad (\theta\in[0,1]) \] [2409.06186]. ## 5. Structural, symbolic, and rigidity viewpoints Homogeneous Moran sets admit several geometric reinterpretations. In the augmented-tree approach, a Moran set \(E\in\mathcal M(J,\{n_k\},\{r_k\})\) is represented by a symbolic graph whose vertices encode approximate scales and whose horizontal edges encode intersections of basic pieces. The resulting augmented tree is hyperbolic, and \(E\) is homeomorphic to its hyperbolic boundary; under the additional separation condition (H), the boundary identification is bi-Hölder [1206.1143]. In the constant-ratio case \(r_k\equiv r\), the symbolic levels simplify to \(X_n=D_n\), and a rearrangeable subclass yields Lipschitz equivalence results [1206.1143]. A different line of work studies homogeneous scaling through microsets and finite clustering. Under the levelwise exponential scaling condition \[ C^{-1}\alpha^{|i|}\le \operatorname{diam}(E_i)\le C\alpha^{|i|} \qquad\text{for all }i\in\Gamma_*, \] together with uniform finite clustering, one has \[ \dim_H(E)=\dim_A(E)=t \] where \(t\) is the pressure zero, and the supremum of microset Hausdorff dimensions agrees with the relevant Assouad quantity [1506.07851]. This suggests that homogeneous Moran scaling can be read either through cylinder combinatorics or through microstructure. In the language of equi-homogeneity, self-similar sets satisfying the Moran open-set condition are equi-homogeneous, and many non-autonomous equal-ratio-at-each-stage pullback attractors are equi-homogeneous as well [1409.4659]. This notion captures the idea that at fixed scales, local covering numbers are comparable across points, while still allowing different dimensional behavior at different scales. Recent quasisymmetric rigidity results isolate special one-dimensional homogeneous Moran classes of Hausdorff dimension \(1\). If \(E\in \mathcal N(I_0,\{n_k\},\{c_k\})\) satisfies the boundary-gap regularity \[ \eta_{\sigma_1,0}=\eta_{\sigma_2,0}=L_{k+1}, \qquad \eta_{\sigma_1,n_k}=\eta_{\sigma_2,n_k}=R_{k+1}, \] together with \[ \dim_H E=1 \] and either \[ \bar\alpha_k\le \omega \underline\alpha_k \qquad\text{or}\qquad \bar\alpha_k\le \theta\, c_1c_2\cdots c_k, \] then every one-dimensional quasisymmetric mapping \(f\) satisfies \[ \dim_H f(E)=1 \] [2510.00540]. Homogeneous perfect sets are included as a special case. ## 6. Arithmetic images and overlap-allowing variants A substantial one-dimensional branch of the subject studies homogeneous Moran sets with overlaps. In the class \((\mathcal M,c_k,n_k,\kappa)\), each level-\(k\) child interval has relative length \(c_k\), each parent has \(n_k\) children, and overlaps are controlled by a uniform parameter \(0\le \kappa<1\) [2005.06163]. The quantity \[ \xi_k=c_k(2+(n_k-2)(1-\kappa)) \] encodes the effective span of a chain of overlapping level-\(k\) intervals [2005.06163]. For \(E_1,E_2\in(\mathcal M,c_k,n_k,\kappa)\) and \(f\in C^3\), the image \[ f(E_1,E_2)=\{f(x,y):x\in E_1,\ y\in E_2\} \] is a closed interval if the signs of \(\partial_x f,\partial_y f\) are globally constant, two directional second-derivative inequalities hold, and \[ 1-\xi_k \le \delta_{xy}\frac{\partial_y f(x,y)}{\partial_x f(x,y)} \le \frac{c_k}{1-\xi_k} \] for all \((x,y)\in[0,1]^2\) and all \(k\ge1\) [2005.06163]. If these conditions hold only from some finite level onward, then \(f(E_1,E_2)\) is a finite union of closed intervals [2005.06163]. A simpler interior criterion appears in the constant-parameter homogeneous subclass \(c_k\equiv c\), \(n_k\equiv n\). If there exists \((x_0,y_0)\in (E_1\times E_2)\cap U\) such that \[ 1-cn < \left| \frac{\partial_y f|_{(x_0,y_0)}}{\partial_x f|_{(x_0,y_0)}} \right| < \frac{c}{1-nc}, \] then \[ f_U(E_1,E_2) \] contains an interior [1905.04645]. In particular, this gives sufficient conditions for sumsets, difference sets, products, and quotients of homogeneous Moran sets to contain intervals or at least nonempty interior. Taken together, these results show that homogeneous Moran sets are not only a convenient subclass for dimension theory. They are also a robust non-autonomous model in which symbolic regularity, metric scaling, arithmetic images, and geometric rigidity remain tractable across Euclidean, overlap-allowing, and complete doubling metric settings [1504.05354][2005.06163][2511.09255]. Source: https://www.emergentmind.com/topics/homogeneous-moran-sets