JULIA SETS WITH AHLFORS-REGULAR CONFORMAL DIMENSION ONE

The paper establishes Theorem A: for a hyperbolic post‑critically finite rational map f, the Ahlfors‑regular conformal dimension of J_f equals 1 if and only if f is a crochet map; see the abstract and statement of Theorem A in the introduction . The proof proceeds via finite subdivision rules, graph virtual endomorphisms, and the Pilgrim–Thurston energy–dimension correspondence (Theorem 6.11: at p* = ARC.dim, the asymptotic p‑energy equals 1) . For crochet maps, Park proves E_1(f) = 1 and E_p(f) < 1 for all p > 1 by exploiting polynomial growth of edge subdivisions and a reduction to isolated Julia vertices and homotopically trivial non‑expanding spines (Section 8) . Conversely, if f is not crochet, the Dudko–Hlushchanka–Schleicher canonical decomposition yields either a Cantor multicurve or a Sierpiński small map ; then monotonicity and lower bounds give E_1(f) > 1, hence ARC.dim(J_f) > 1 (Corollary 7.8) . The candidate model’s solution follows the same blueprint: it invokes the energy–dimension dictionary, shows polynomial vs exponential combinatorics yield E_1 ≤ 1 (< 1 for p > 1) vs E_1 > 1, and concludes ARC.dim(J_f) = 1 iff f is crochet. Minor imprecision: the model writes E_1 ≤ 1 instead of the paper’s sharper E_1 = 1 for crochet maps, but its conclusion relies on the same mechanism and is consistent with Park’s equality result.