Branching-Process RIFS

• Branching-Process RIFS is a random fractal model that integrates Galton–Watson branching with iterated contraction maps to generate nested and multifractal structures.
• The construction uses strict nestedness and leaf/subtree duality, ensuring each new subtree is embedded in a strictly smaller interval than its parent.
• Under mild non-degeneracy assumptions, the model rigorously yields multifractality with distinct behaviors in both non-anchored and anchored cases.

Searching arXiv for the cited work and closely related RIFS/branching references. arXiv search query: "(Hudnall, 30 Sep 2025) Branching-Process Random Iterated Function System multifractality" A branching-process random iterated function system (RIFS) is a random fractal construction that combines Galton–Watson branching processes with random iterated function system theory by requiring that each leaf of a branching tree generate a new subtree embedded at a strictly smaller geometric scale. In the formulation introduced in "Multifractality in the Tree of Life: A Branching-Process RIFS Proof" (Hudnall, 30 Sep 2025), the ambient space is the unit interval $X=[0,1]$, the root is identified with the whole interval, and every subsequent replacement step couples stochastic reproduction with stochastic contraction. The resulting object is neither merely a branching tree with labels nor merely a random geometric IFS; it is a branching tree whose leaves each generate a new, randomly contracted copy of the process. Under explicit mild assumptions, this framework yields a rigorous multifractality theorem in both a non-anchored case with a nontrivial compact attractor and an anchored case in which the invariant set collapses to a point while tangent measures retain the same multifractal law (Hudnall, 30 Sep 2025).

1. Definition and recursive construction

The defining mechanism is recursive. At depth $k=0$, the root satisfies

$I(\mathrm{root})=X,\qquad s(\mathrm{root})=1.$

For each step $k\ge 1$, every leaf $v$ of the current tree is replaced by a finite Galton–Watson subtree $\mathcal{G}_{k,v}$, and that subtree is embedded into a subinterval $I_{k,v}\subseteq I(v)$ whose diameter is strictly smaller than that of its parent. The contraction factor is random: $$R_{k,v}\sim \nu_{s(v)} \quad\text{with support in }(0,s(v)),$$ where $s(v)=\operatorname{diam}(I(v))$. The key recursive property is

$R_{k,v}<s(v)\quad\text{a.s.}$

so every new subtree lives at a strictly smaller scale than the leaf that generates it. In similarity-map form, each embedded subtree is represented by

$k=0$0

The branching component is specified by a Galton–Watson offspring law

$k=0$1

with finite support. The geometric component is specified by the recursively contracted maps $k=0$2. The unification is therefore structural rather than superficial: branching supplies the genealogical dependence, while the random contractions supply recursive geometric nesting. A plausible implication is that the natural state space of the model is simultaneously genealogical and geometric, with the geometry indexed by the evolving branching tree rather than by a fixed symbolic code.

2. Nestedness, duality, and randomness

The model is built to encode three principles identified explicitly in the source formulation: nestedness, duality, and randomness (Hudnall, 30 Sep 2025). Nestedness is expressed by strict containment along any root-to-leaf path,

$k=0$3

Because each contraction factor is supported in $k=0$4, child intervals are strictly smaller than parent intervals, so the recursive geometry is genuinely nested rather than merely self-similar in distribution.

Duality is the leaf/subtree dual role of each node. Relative to its parent, a node is a leaf; relative to its descendants, it is the root of a new subtree. In the paper’s terminology this encodes whole/part duality: an entity is both a complete unit and part of a larger system. Randomness enters in two independent ways: through the offspring number $k=0$5 and through the contraction factors $k=0$6. These random variables are independent across leaves and independent of each other, so both genealogical structure and geometric scaling fluctuate.

This combination distinguishes the construction from standard random-iteration models in which a single particle follows one random orbit. The survey "Iterated Function Systems: A Comprehensive Survey" emphasizes random map selection, Markov operators, invariant measures, and random trajectories of the form $k=0$7 (Ghosh et al., 2022). By contrast, the branching-process RIFS proliferates a hierarchy of nested supports because every leaf spawns a new randomly contracted copy of the process. This suggests that its natural observables are tree-indexed measures and level-set spectra rather than only trajectory-wise ergodic averages.

3. Assumptions and structural variants

The main theorem is proved under two layers of assumptions: biological simplifications and mathematical non-degeneracy (Hudnall, 30 Sep 2025). The biological simplifications are strictly branching reproduction, no merging, reproduction once at the end of lifetime, and finite-support offspring distribution. The paper presents these as simplifications rather than as the essential source of the theorem.

The key mathematical assumptions are more restrictive. The offspring distribution is not concentrated on $k=0$8 or $k=0$9, so there is positive probability of at least two offspring. Offspring numbers and contraction factors are independent, both across leaves and from one another. Contraction factors are continuous random variables supported on $I(\mathrm{root})=X,\qquad s(\mathrm{root})=1.$0, and their distribution is non-degenerate, meaning not concentrated at a single value. Because the offspring law has finite support, the number of children is uniformly bounded.

Under these assumptions, the paper states that the attractor is uncountable almost surely, the limiting measure is nontrivial, and the $I(\mathrm{root})=X,\qquad s(\mathrm{root})=1.$1-spectrum is strictly convex rather than collapsing to a point. Two variants are then distinguished. In Variant A, the non-anchored case, the system produces a nontrivial compact attractor. In Variant B, the anchored case, one fixes the translation term by

$I(\mathrm{root})=X,\qquad s(\mathrm{root})=1.$2

so every chain converges to $I(\mathrm{root})=X,\qquad s(\mathrm{root})=1.$3 and the geometric invariant set collapses to the singleton $I(\mathrm{root})=X,\qquad s(\mathrm{root})=1.$4. A common misconception is that such collapse eliminates the multifractal content of the model. The theorem says otherwise: in the anchored case the invariant set degenerates, but tangent measures obey the same multifractal law as in the non-anchored case (Hudnall, 30 Sep 2025).

4. Measures, spectra, and multifractal formalism

The depth-$I(\mathrm{root})=X,\qquad s(\mathrm{root})=1.$5 construction carries a natural cascade measure $I(\mathrm{root})=X,\qquad s(\mathrm{root})=1.$6, with mass assigned by products of contraction factors along root-to-leaf paths (Hudnall, 30 Sep 2025). For the leaf intervals $I(\mathrm{root})=X,\qquad s(\mathrm{root})=1.$7 at depth $I(\mathrm{root})=X,\qquad s(\mathrm{root})=1.$8, the partition function is

$I(\mathrm{root})=X,\qquad s(\mathrm{root})=1.$9

A representative mesh scale is

$k\ge 1$0

and the $k\ge 1$1-spectrum is defined by

$k\ge 1$2

The paper also writes

$k\ge 1$3

The multifractal formalism is expressed through local scaling exponents. The level sets are

$k\ge 1$4

and on the relevant interval the paper proves

$k\ge 1$5

In the non-anchored case, the measures $k\ge 1$6 converge in distribution to a limiting probability measure $k\ge 1$7 supported on a compact, totally disconnected, uncountable, Cantor-like attractor $k\ge 1$8. Its $k\ge 1$9-spectrum exists almost surely and is strictly convex, and its Legendre transform gives a nontrivial multifractal spectrum. In the anchored case, the invariant geometric set is $v$0, but if $v$1 is a leaf at depth $v$2 and $v$3 rescales $v$4 to $v$5, then the rescaled measures

$v$6

have weak limits with the same law as the non-anchored limiting measure.

5. Proof architecture

The proof is organized through a sequence of lemmas and a multifractal formalism theorem (Hudnall, 30 Sep 2025). First, the depth-$v$7 measures $v$8 form a martingale and converge almost surely to a limiting random probability measure $v$9. The ingredients cited are shrinking diameters of nested intervals, Cantor’s intersection theorem, and Kahane–Peyrière martingale convergence. This yields the Cantor-like attractor in the non-anchored case.

Second, the existence of the $\mathcal{G}_{k,v}$0-spectrum follows from subadditivity and Kingman’s subadditive ergodic theorem: $\mathcal{G}_{k,v}$1 Third, convexity and strict convexity of $\mathcal{G}_{k,v}$2 are obtained from the convexity of the log-sum-exp expression $\mathcal{G}_{k,v}$3. Strict convexity follows from non-degeneracy of the contraction law; if the contraction distribution is not a point mass, then the spectrum does not collapse. The paper identifies an interval $\mathcal{G}_{k,v}$4 on which $\mathcal{G}_{k,v}$5 is finite and differentiable, with $\mathcal{G}_{k,v}$6 and

$\mathcal{G}_{k,v}$7

The multifractal formalism then uses a quasi-Bernoulli / weak Gibbs property and a standard two-sided argument. The upper bound controls the number of cylinders with a given scaling exponent through partition sums, yielding

$\mathcal{G}_{k,v}$8

The lower bound tilts the measure via

$\mathcal{G}_{k,v}$9

extracts a weak limit $I_{k,v}\subseteq I(v)$0, proves that $I_{k,v}\subseteq I(v)$1 is supported on $I_{k,v}\subseteq I(v)$2, and then applies a Frostman argument. In the anchored case, the central regeneration step is that the rescaled subtree below any leaf has the same law as the non-anchored process after affine normalization. Thus tangent measures recover the same $I_{k,v}\subseteq I(v)$3 and the same multifractal spectrum.

6. Position within the RIFS and branching literature

The acronym RIFS is not uniform across the literature. In "Hidden variable recurrent fractal interpolation function with four function contractivity factors" (Yun, 2019), RIFS denotes recurrent iterated function systems used for fractal interpolation. There the emphasis is on local self-similarity, domain-to-region recurrence encoded by a connection matrix, Read–Bajraktarević operators, and hidden-variable interpolation surfaces and curves. That framework is not a probabilistic branching-process model. By contrast, the branching-process RIFS is explicitly probabilistic and built from Galton–Watson reproduction coupled to random contractions.

The broader IFS survey literature treats random iteration primarily through Markov evolution, invariant measures, and state-dependent or state-independent random map selection (Ghosh et al., 2022). The branching-process RIFS instead attaches a new random geometric copy of the process to every leaf, so its recursion is tree-valued rather than orbit-valued. A nearby branching-cascade construction appears in "A class of multifractal processes constructed using an embedded branching process" (Decrouez et al., 2012), where a crossing tree, multiplicative cascade weights, and a Galton–Watson genealogy generate cascade measures on tree boundaries. That model is structurally related, but it is not the same object as the interval-based branching-process RIFS of (Hudnall, 30 Sep 2025). Likewise, "Scaling limits in dependent random environments: relating a random walk, a branching process and a spatial branching process" develops recursive branching structure through composed generating functions and genealogical contour dynamics, but does not present a formal branching-process RIFS theory (Buchanan, 13 Dec 2025).

Within this landscape, the distinctive contribution of the branching-process RIFS is the theorem that multifractality follows from the combined presence of strict nestedness, leaf/subtree duality, and independent randomness under mild non-degeneracy assumptions. In the non-anchored case, multifractality belongs to the invariant measure on a Cantor-like attractor. In the anchored case, it survives at the level of tangent measures even though the invariant geometric set degenerates to a point. This suggests a minimal-condition theorem: multifractality is not imposed as an auxiliary hypothesis but emerges from the recursive branching geometry itself (Hudnall, 30 Sep 2025).