When do two rational functions have locally biholomorphic Julia sets?

The paper’s Theorem A precisely states and proves that if σ locally preserves the maximal-entropy measure class (with the smooth-J case handled by proportionality) and maps a repelling periodic point of f1 to a preperiodic point of f2, then there are a,b and an irreducible algebraic curve Z ⊂ P^1×P^1 preperiodic under (f1^a,f2^b) that contains the graph of σ; moreover d1^a = d2^b and σ▹μ_{f2} ∝ μ_{f1} follow as corollaries . The proof constructs a generalized Poincaré–Koenigs parametrization and shows Z is F-invariant for F=(f1^a,f2^b), then compares the induced entropy measures via the projections to deduce the stated consequences . By contrast, the candidate solution makes a pivotal but unjustified leap: from “Z contains an open piece of the graph of σ” it concludes deg(π1|Z)=1, hence Z is the global graph of a rational map σ~, yielding a global semiconjugacy f2^b∘σ~=σ~∘f1^a. This inference is not warranted; an irreducible algebraic curve can project with degree >1 even if it contains a local analytic graph branch over an open set. The paper does not claim (nor need) that Z is a graph; instead it derives the degree equality and measure proportionality from F-invariance and entropy considerations on Z . The model’s end conclusions agree with the paper, but its proof hinges on an incorrect step about the projection degree, so the model’s argument is flawed.