Bounded Assouad Characteristic Overview

• Bounded Assouad characteristic is a collective term defining controlled worst-case scaling behavior through various Assouad-type invariants such as the Assouad dimension, lower dimension, and spectra.
• It unifies several formal approaches by capturing uniform upper and lower covering growth, dilation control functions, and neighborhood thickness via scale-invariant inequalities.
• This concept supports precise structural theorems and transformation laws in fractal geometry, dynamical systems, and quasiregular mappings, offering actionable insights into scaling properties.

Searching arXiv for the exact phrase and closely related Assouad-type notions to ground the article in current literature. “Bounded Assouad characteristic” is not a standard named invariant in the cited Assouad-dimension literature. In several relevant arXiv treatments, the phrase is explicitly absent, and the formal content is carried instead by a family of Assouad-type quantities and witnesses: the Assouad dimension $\dim_A$, lower dimension $\dim_L$, Assouad spectrum, Assouad–Nagata dimension $\ANdim$ via dilation control functions, and upper Assouad codimension via Aikawa-type integral conditions (Distel, 2023, Fraser et al., 2022, Garitsis, 2023). In that sense, the phrase is best interpreted as a collective label for bounded or controlled worst-case scaling behavior rather than as a single universally fixed definition.

1. Terminological status and nearest formal notions

A potential source of confusion is the expectation that “bounded Assouad characteristic” denotes one canonical scalar invariant. The cited papers do not support that usage. The graph-theoretic paper on Assouad–Nagata dimension states that it “does not introduce a separate named invariant called a ‘bounded Assouad characteristic’,” and identifies the closest formal objects as dilation control functions, $\ell$-almost control functions, $(k,\ell)$-centred sets, $(k,\ell)$-partitions, and $(k,\ell,\mathcal H)$-strong-constructions (Distel, 2023). The paper on parabolic Julia sets likewise states that it does not define or use a notion called “bounded Assouad characteristic,” and instead develops the Assouad dimension, lower dimension, Assouad spectrum, lower spectrum, and the corresponding notions for the $h$-conformal measure (Fraser et al., 2022). The planar quasiregular-distortion paper makes the same point, and treats “bounded/controlled Assouad-type behavior” through the Assouad dimension, the regularized Assouad spectrum, and porosity (Garitsis, 2023).

The nearest formal notions therefore split into several regimes. At small scales, $\dim_A$, $\dim_L$, and their spectra quantify uniform upper and lower covering growth. In large-scale graph geometry, $\dim_L$0 expresses boundedness through the existence of a dilation control function. In codimension theory, upper Assouad codimension measures neighborhood thickness from below and is characterized by an upper Aikawa condition. This suggests that “bounded Assouad characteristic” is most accurately treated as an umbrella expression for several inequivalent but tightly related witnesses of controlled scaling.

2. Core Assouad-type dimensions and spectra

For a subset $\dim_L$1 of a metric space $\dim_L$2, the Assouad dimension is defined by

$\dim_L$3

A global version is also available: $\dim_L$4 If $\dim_L$5 is bounded, the local and global definitions coincide, and a metric space $\dim_L$6 is doubling iff $\dim_L$7 (Chen et al., 2016).

The lower dimension is the dual quantity

$\dim_L$8

Thus $\dim_L$9 records a uniform upper scaling rate, whereas $\ANdim$0 records a uniform lower scaling rate (Chen et al., 2016). For compact sets, the standard chain

$\ANdim$1

is emphasized in the Julia-set setting (Fraser et al., 2022).

A spectral refinement imposes a scale relation between $\ANdim$2 and $\ANdim$3. For $\ANdim$4, one formulation is

$\ANdim$5

while the upper spectrum allows $\ANdim$6: $\ANdim$7 The generalized upper box dimension is then defined by

$\ANdim$8

and for bounded $\ANdim$9 one has $\ell$0 (Wang et al., 1 Oct 2025).

3. Boundedness via control functions and codimension

In graph classes, bounded Assouad-type behavior is formalized by Assouad–Nagata dimension. For a weighted graph $\ell$1, an $\ell$2-dimensional control function $\ell$3 requires that for every $\ell$4 there exist $\ell$5 collections $\ell$6 covering $\ell$7, each $\ell$8 being $\ell$9-disjoint, and every $(k,\ell)$0 satisfying

$(k,\ell)$1

A function $(k,\ell)$2 is a dilation if $(k,\ell)$3 for all $(k,\ell)$4, and $(k,\ell)$5 is the least $(k,\ell)$6 for which $(k,\ell)$7 admits an $(k,\ell)$8-dimensional control function that is also a dilation (Distel, 2023). The paper proves that for every proper minor-closed class $(k,\ell)$9,

$(k,\ell)$0

and that $(k,\ell)$1 iff $(k,\ell)$2 has bounded treewidth. For subdivision-closed classes, bounded Assouad–Nagata dimension is equivalent to excluding some fixed minor (Distel, 2023). In this setting, a “bounded Assouad characteristic” is naturally represented by the existence of a dilation control function.

A different boundedness formalism appears in upper Assouad codimension. In a doubling metric measure space $(k,\ell)$3, for a nonempty set $(k,\ell)$4, the upper Assouad codimension is the infimum of all $(k,\ell)$5 such that there exists $(k,\ell)$6 with

$(k,\ell)$7

for every $(k,\ell)$8 and all $(k,\ell)$9 (Kline et al., 2023). The same paper introduces the upper Aikawa condition: for $(k,\ell,\mathcal H)$0, $(k,\ell,\mathcal H)$1 satisfies it if for every $(k,\ell,\mathcal H)$2 there exists $(k,\ell,\mathcal H)$3 such that

$(k,\ell,\mathcal H)$4

whenever $(k,\ell,\mathcal H)$5, $(k,\ell,\mathcal H)$6, and $(k,\ell,\mathcal H)$7 is Borel with $(k,\ell,\mathcal H)$8. The main equivalence is that the upper Assouad codimension is the infimum of all $(k,\ell,\mathcal H)$9 for which the upper Aikawa condition holds (Kline et al., 2023). This converts a neighborhood-thickness bound into an integral criterion and connects the codimension to a local fractional Hardy inequality.

4. Accessibility under subsets and monotonicity under limits

One of the strongest subset-stability results is the accessibility theorem for $h$0 and $h$1. If $h$2 is a subset of a doubling metric space, then for every

$h$3

there exists $h$4 such that

$h$5

and similarly, for every

$h$6

there exists $h$7 such that

$h$8

This is Theorem 1 of (Chen et al., 2016). The proof reduces to Euclidean space by snowflaking $h$9, applying the Assouad embedding theorem, and using the scaling identities

$\dim_A$0

In Euclidean space the argument passes through the star dimension $\dim_A$1, defined from $\dim_A$2-adic cube counts by

$\dim_A$3

with $\dim_A$4 for $\dim_A$5 (Chen et al., 2016).

Complementing accessibility under subsets is monotonicity under Gromov–Hausdorff limit constructions. The pseudo-cone framework generalizes both tangent cones and asymptotic cones, and if $\dim_A$6 is a pseudo-cone of a metric space $\dim_A$7, then

$\dim_A$8

The same paper gives lower- and conformal-Assouad analogues: $\dim_A$9 for the relevant hypotheses (Ishiki, 2019). This suggests that bounded Assouad-type data are simultaneously flexible under taking subsets and monotone under pseudo-cone limit formation.

5. Explicit evaluations in fractal and dynamical models

Concrete calculations show what these boundedness notions detect. For Bedford–McMullen carpets with $\dim_L$0, if $\dim_L$1 is the number of occupied rows and $\dim_L$2 is the maximal number of rectangles in a row, then

$\dim_L$3

For Lalley–Gatzouras carpets, with $\dim_L$4 the dimension of the vertical Cantor projection and $\dim_L$5 the maximal Hausdorff dimension of the horizontal fibers,

$\dim_L$6

In both families the proofs rely on approximate squares for the upper bound and weak tangents containing product sets $\dim_L$7 for the lower bound (Mackay, 2010). The same paper gives a sharp dichotomy for conformal Assouad dimension: it is either $\dim_L$8 or $\dim_L$9, depending on whether a tangent contains an interval factor or the carpet is uniformly disconnected (Mackay, 2010).

In holomorphic dynamics, for a parabolic Julia set $\dim_L$00 with Hausdorff dimension $\dim_L$01,

$\dim_L$02

The full Assouad spectrum is computed: $\dim_L$03 and the lower spectrum has the dual form

$\dim_L$04

For the associated $\dim_L$05-conformal measure $\dim_L$06,

$\dim_L$07

If the Julia set has a Cremer point, then $\dim_L$08 (Fraser et al., 2022). These formulas show that bounded Assouad-type behavior can differ sharply from Hausdorff, box, and packing dimensions even in classical dynamical systems.

6. Distortion, porosity, and extremal spectrum behavior

Assouad-type boundedness is also meaningful under mappings. If $\dim_L$09 is a non-constant $\dim_L$10-quasiregular map and $\dim_L$11 is compact with $\dim_L$12, then

$\dim_L$13

For every $\dim_L$14, the paper also proves a spectrum distortion bound of the same Astala-type form, with $\dim_L$15 replaced by the spectrum value $\dim_L$16 and the corresponding $\dim_L$17-normalization (Garitsis, 2023). A key ingredient is the holomorphic monotonicity theorem: if $\dim_L$18 is non-constant and holomorphic and $\dim_L$19 is compact, then

$\dim_L$20

Since porosity of compact subsets of $\dim_L$21 is characterized by $\dim_L$22, the distortion theorem yields invariance of porosity under planar quasiregular maps (Garitsis, 2023).

At the spectral endpoints, the generalized upper box framework gives further rigidity. For arbitrary $\dim_L$23,

$\dim_L$24

and the paper proves

$\dim_L$25

It also shows that for every $\dim_L$26,

$\dim_L$27

(Wang et al., 1 Oct 2025). This isolates the zero and full-dimension regimes as rigid extremal manifestations of bounded Assouad-type behavior.

Taken together, these results indicate that “bounded Assouad characteristic” is most usefully understood as a family of quantitative witnesses for controlled covering growth, neighborhood thickness, or large-scale decomposition. The formal representatives of that family are not interchangeable: $\dim_L$28, $\dim_L$29, spectra, $\dim_L$30, and upper Assouad codimension measure different aspects of uniform scaling. What unifies them is that each expresses boundedness through scale-invariant inequalities, and each supports precise structural theorems, exact computations, or sharp transformation laws in its natural setting.