Homogeneous Moran Sets

• Homogeneous Moran sets are fractal constructions with uniform offspring numbers and contraction ratios at each level.
• They enable explicit computations of Hausdorff, packing, and intermediate dimensions using cumulative branching and contraction ratios.
• The model adapts to various settings including Euclidean intervals, overlap-allowed constructions, and complete doubling metric spaces.

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 $k$ (Käenmäki et al., 2015). 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 (Du et al., 2024, Li et al., 2024, Li et al., 2020). 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 (Käenmäki et al., 2015, Miao et al., 12 Nov 2025, Li et al., 1 Oct 2025).

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 $N_k$0, nested inside their parents, and have pairwise disjoint interiors at each fixed level (Du et al., 2024, Li et al., 2014). The homogeneous specialization is the condition

$N_k$1

so every level-$N_k$2 child has diameter $N_k$3 times the diameter of its parent (Du et al., 2024). In this setting the total number and common diameter of level-$N_k$4 basic sets are

$N_k$5

in the notation of one-dimensional interval models (Li et al., 2024).

A more general metric-space formulation replaces Euclidean geometry by a complete doubling metric space $N_k$6, uses a codetree $N_k$7, and defines a Moran construction by nested compact sets $N_k$8 with diameters tending to zero along branches, disjoint offspring, a uniform interior ball condition, and asymptotically negligible diameter fluctuations (Käenmäki et al., 2015). In that framework, a Moran construction is asymptotically homogeneous when it is asymptotically spatially symmetric and

$N_k$9

This is the exact analogue of the standard level-homogeneous Moran construction, but now valid in complete doubling metric spaces (Käenmäki et al., 2015).

One-dimensional interval models often make the level data completely explicit. In the disjoint setting, a homogeneous Moran structure on $n_k$0 is a family of closed intervals $n_k$1 such that each level-$n_k$2 interval contains $n_k$3 children with pairwise disjoint interiors and

$n_k$4

(Li et al., 2024). In the overlap-allowing model $n_k$5, the children still have common relative length $n_k$6, but overlaps are allowed with

$n_k$7

and the intersection of two level-$n_k$8 basic intervals is controlled relative to their length (Li et al., 2020). 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 $n_k$9 is defined by

$c_k$0

In the homogeneous case this becomes

$c_k$1

and the dimension formulas simplify to

$c_k$2

$c_k$3

In the stationary homogeneous case $c_k$4, $c_k$5, these reduce to

$c_k$6

(Käenmäki et al., 2015).

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 $c_k$7 for all $c_k$8, and the pressure-zero equation becomes

$c_k$9

If $N_k$0, then

$N_k$1

which is the same classical homogeneous Moran formula (Holland et al., 2012). In the Moran-type IFS framework, the homogeneous specialization $N_k$2 likewise yields

$N_k$3

(Cao et al., 16 Jan 2026).

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 $N_k$4, $N_k$5, one defines

$N_k$6

and under one of three interior-gap conditions obtains

$N_k$7

(Li et al., 1 Oct 2025). This shows that once boundary trimming is built into the model, the effective scale may be $N_k$8 rather than $N_k$9.

3. Measures, local dimensions, and $c_k$0-dimensions

For measures on Moran constructions, local dimension can be read directly from nested cylinders. If $c_k$1 is supported on the limit set $c_k$2 of a Moran construction in a complete doubling metric space, then for $c_k$3-almost all $c_k$4,

$c_k$5

$c_k$6

(Käenmäki et al., 2015).

The canonical measure in the homogeneous case is the uniformly distributed measure

$c_k$7

For an asymptotically homogeneous Moran construction, this measure has constant local and global $c_k$8-dimensions: $c_k$9

$k$0

for all $k$1 (Käenmäki et al., 2015). If the ratio

$k$2

actually converges, then for all $k$3,

$k$4

This is the strongest monofractal-type statement in the paper (Käenmäki et al., 2015).

For more general Moran measures, the same paper proves an entropy-average theorem under a growth assumption on $k$5 and an $k$6-type summability condition. This gives almost-sure local dimension formulas in terms of averages of $k$7 and $k$8, but the uniformly distributed homogeneous case is the cleanest closed-form specialization (Käenmäki et al., 2015).

4. Assouad-type and intermediate dimensions

For one-dimensional homogeneous Moran sets with bounded branching,

$k$9

the Assouad dimension is

$J\subset \mathbb{R}^d$0

and the paper proves the upper bound

$J\subset \mathbb{R}^d$1

for the lower dimension (Li et al., 2024). The interpretation is direct: the Assouad dimension records the densest asymptotic level window, while the lower dimension is controlled by the sparsest one.

For quasi-Assouad dimension, the modern general theory introduces index sets selecting windows whose cumulative contraction is not too small relative to the preceding scale. In the homogeneous case, under

$J\subset \mathbb{R}^d$2

one obtains an exact formula

$J\subset \mathbb{R}^d$3

because all homogeneous Moran structures are quasi-normal (Miao et al., 12 Nov 2025). If, in addition,

$J\subset \mathbb{R}^d$4

and BBC holds, then

$J\subset \mathbb{R}^d$5

(Miao et al., 12 Nov 2025).

Intermediate dimensions furnish a different interpolation between Hausdorff and box-counting behavior. For homogeneous Moran sets, define $J\subset \mathbb{R}^d$6 by

$J\subset \mathbb{R}^d$7

Then, when $J\subset \mathbb{R}^d$8,

$J\subset \mathbb{R}^d$9

$D_k$0

(Du et al., 2024). These formulas show that even homogeneous Moran sets need not have a genuine intermediate dimension: one explicit example has $D_k$1 and branching numbers alternating in huge factorial blocks between $D_k$2 and $D_k$3, with

$D_k$4

(Du et al., 2024).

5. Structural, symbolic, and rigidity viewpoints

Homogeneous Moran sets admit several geometric reinterpretations. In the augmented-tree approach, a Moran set $D_k$5 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 $D_k$6 is homeomorphic to its hyperbolic boundary; under the additional separation condition (H), the boundary identification is bi-Hölder (Luo, 2012). In the constant-ratio case $D_k$7, the symbolic levels simplify to $D_k$8, and a rearrangeable subclass yields Lipschitz equivalence results (Luo, 2012).

A different line of work studies homogeneous scaling through microsets and finite clustering. Under the levelwise exponential scaling condition

$D_k$9

together with uniform finite clustering, one has

$J_u$0

where $J_u$1 is the pressure zero, and the supremum of microset Hausdorff dimensions agrees with the relevant Assouad quantity (Käenmäki et al., 2015). 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 (Henderson et al., 2014). 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 $J_u$2. If $J_u$3 satisfies the boundary-gap regularity

$J_u$4

together with

$J_u$5

and either

$J_u$6

then every one-dimensional quasisymmetric mapping $J_u$7 satisfies

$J_u$8

(Li et al., 1 Oct 2025). 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 $J_u$9, each level-$N_k$00 child interval has relative length $N_k$01, each parent has $N_k$02 children, and overlaps are controlled by a uniform parameter $N_k$03 (Li et al., 2020). The quantity

$N_k$04

encodes the effective span of a chain of overlapping level-$N_k$05 intervals (Li et al., 2020).

For $N_k$06 and $N_k$07, the image

$N_k$08

is a closed interval if the signs of $N_k$09 are globally constant, two directional second-derivative inequalities hold, and

$N_k$10

for all $N_k$11 and all $N_k$12 (Li et al., 2020). If these conditions hold only from some finite level onward, then $N_k$13 is a finite union of closed intervals (Li et al., 2020).

A simpler interior criterion appears in the constant-parameter homogeneous subclass $N_k$14, $N_k$15. If there exists $N_k$16 such that

$N_k$17

then

$N_k$18

contains an interior (Ren et al., 2019). 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 (Käenmäki et al., 2015, Li et al., 2020, Miao et al., 12 Nov 2025).