Multifractal Analysis of F-Exponents for Finitely Irreducible Conformal Graph Directed Markov Systems

The paper’s main theorem (existence/uniqueness and real-analyticity of (t(ξ),q(ξ)) solving P(t,q)=qξ and ∂P/∂q=ξ; and the equalities HD(J(ξ))=HD(J1(ξ))=t(ξ)) is stated as Theorem 1.1.1 and proved in Section 4 via Proposition 4.1.1 (dimension bounds using the volume lemma) and Proposition 4.1.3 (existence/uniqueness/analyticity via convexity and the implicit function theorem), together with the derivative formulas (Lemma 2.2.1) and structure of the Manhattan region (Lemmas 2.1.2, 2.1.4) and boundary behavior (Lemma 2.2.5) . The model’s solution follows the same route: pressure regularity and derivatives, convexity/strict monotonicity, a Legendre-type minimization W(t,ξ)=P(t,q)-ξq with W_t<0 to locate t(ξ), and IFT for analyticity, then Gibbs/volume-lemma lower bound and Carathéodory-type upper bound; it also treats the comparability vs bounded F cases exactly as in the paper. Minor stylistic differences (e.g., invoking variance/Hessian identities) do not affect correctness. We find no substantive gaps: the paper ensures non-degeneracy for the IFT via strict convexity (ft,q not cohomologous to a constant) and uses SOSC to handle overlaps (hence µ(J>1)=0) .