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Multifractality in the Tree of Life: A Branching-Process RIFS Proof

Kevin Hudnall

TL;DR

The paper addresses why the tree of life exhibits multifractal scaling by introducing a branching-process random iterated function system (RIFS) that couples Galton–Watson branching with recursive contractions. It develops a rigorous multifractal formalism for this genealogically dynamic fractal, proving the existence and properties of an $L^q$-spectrum $ au(q)$ and a Legendre-derived spectrum $f( obreak obreak ( obreak))$ under mild non-degeneracy assumptions. It analyzes two variants: a non-anchored case with a Cantor-like attractor and an anchored case where the invariant set collapses to a point, yet tangent measures recover the same multifractal law, highlighting mortality as a biologically natural constraint. The results provide a principled explanation for the ubiquity of multifractal signatures in biological data and extend the multiplicative-cascade framework to a genealogically recursive setting, with potential broad implications for modeling evolution and scaling in biological systems.

Abstract

We study a branching-process random iterated function system (RIFS) that formalizes the foundational principles of nestedness, duality, and randomness in the living tree of life (Hudnall & D'Souza, 2025). In this construction, each leaf of a branching process generates a subtree at a strictly smaller contraction scale, thereby unifying classical branching processes and random IFS theory in a single framework. We prove rigorously that this branching-process RIFS is multifractal under explicit, mild assumptions. Two variants are analyzed: a non-anchored case with a nontrivial compact attractor, and a biologically motivated anchored case in which the invariant set collapses to a point while tangent measures obey the same multifractal law. Thus, multifractality emerges as a necessary mathematical consequence of nestedness, duality, and randomness, yielding a minimal-condition theorem that explains the ubiquity of multifractal signatures in biological data.

Multifractality in the Tree of Life: A Branching-Process RIFS Proof

TL;DR

The paper addresses why the tree of life exhibits multifractal scaling by introducing a branching-process random iterated function system (RIFS) that couples Galton–Watson branching with recursive contractions. It develops a rigorous multifractal formalism for this genealogically dynamic fractal, proving the existence and properties of an -spectrum and a Legendre-derived spectrum under mild non-degeneracy assumptions. It analyzes two variants: a non-anchored case with a Cantor-like attractor and an anchored case where the invariant set collapses to a point, yet tangent measures recover the same multifractal law, highlighting mortality as a biologically natural constraint. The results provide a principled explanation for the ubiquity of multifractal signatures in biological data and extend the multiplicative-cascade framework to a genealogically recursive setting, with potential broad implications for modeling evolution and scaling in biological systems.

Abstract

We study a branching-process random iterated function system (RIFS) that formalizes the foundational principles of nestedness, duality, and randomness in the living tree of life (Hudnall & D'Souza, 2025). In this construction, each leaf of a branching process generates a subtree at a strictly smaller contraction scale, thereby unifying classical branching processes and random IFS theory in a single framework. We prove rigorously that this branching-process RIFS is multifractal under explicit, mild assumptions. Two variants are analyzed: a non-anchored case with a nontrivial compact attractor, and a biologically motivated anchored case in which the invariant set collapses to a point while tangent measures obey the same multifractal law. Thus, multifractality emerges as a necessary mathematical consequence of nestedness, duality, and randomness, yielding a minimal-condition theorem that explains the ubiquity of multifractal signatures in biological data.

Paper Structure

This paper contains 39 sections, 8 theorems, 59 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Main result
  3. Model definition
  4. Nestedness, duality, and randomness
  5. Assumptions and biological restrictions
  6. Multifractality of the branching-process RIFS
  7. Proof sketch.
  8. Existence of the limiting measure and attractor structure
  9. Pressure and $L^q$-spectrum
  10. Anchored case: tangent measures and multifractality
  11. Summary
  12. Numerical illustration
  13. Discussion
  14. Conclusion
  15. Additional proofs
  16. ...and 24 more sections

Key Result

Theorem 1

Consider the branching-process RIFS under the assumptions of Section sec:model. In Variant A (non-anchored), the sequence of measures $\{\mu_n\}$ converges in distribution to a limiting measure $\mu$ supported on a Cantor-like attractor in $[0,1]$. The associated $L^q$-spectrum exists almost surely, is strictly convex in $q$, and its Legendre transform yields a nontrivial multifractal spectrum $f

Figures (1)

  • Figure 1: Numerical illustration of the branching-process RIFS. Top: Distribution of interval masses across normalized position and iteration depth (logarithmic scale; warmer colors indicate greater mass). Each row corresponds to a generation, showing the recursive embedding of lineages into smaller scales. Bottom: Estimated multifractal spectrum $f(\alpha)$, obtained from partition sums of the scale matrix and Legendre transform of the scaling exponents. The strictly convex curve confirms the multifractal law of Theorem \ref{['thm:formalism']}, showing the coexistence of multiple scaling exponents within a single realization.

Theorems & Definitions (22)

  • Theorem 1: Multifractality of the branching-process RIFS
  • Lemma 2
  • proof
  • Lemma 3: Existence of the $L^q$-spectrum
  • proof
  • Lemma 4: Domain and convexity of $\tau$
  • proof
  • Definition 2: Multifractal spectrum
  • Corollary 5: Nontrivial multifractal spectrum
  • Theorem 6: Multifractal formalism on an interval
  • ...and 12 more