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

Abstract

We study a branching-process random iterated function system (RIFS) defined by a recursive replacement of leaves by finite subtrees at strictly smaller contraction scales. This construction yields a tree-valued, infinite-depth random geometry that unifies classical branching processes and random iterated function systems while remaining distinct from both. We prove rigorously that the resulting branching-process RIFS is multifractal under explicit and 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 geometric set collapses to a point, while tangent measures obey the same multifractal law. The construction formalizes the foundational principles of nestedness, duality, and randomness in the living tree of life (Hudnall & D'Souza, 2025), yielding a minimal-condition theorem that explains the ubiquity of multifractal signatures in biological data.