The Hausdorff dimension of Julia sets of meromorphic functions in the Speiser class

The paper proves that for every d in (0,2] there exists a meromorphic f with three singularities of the inverse and dim J(f)=d (stated already in the abstract and introduction) . Its construction uses an elliptic G with critical values 0,1,∞, an explicit example with J=Ĉ for d=2, a family H=ηG^p in S3, and then hm(z)=H(m arcsin(z/m)), followed by the scaled family fλ(z)=hm(λz). It establishes limλ→0 dim J(fλ)=0, a uniform lower bound dim J(hm)>8p/(4p+1) for large m (hence near 2 for large p), and continuity of λ↦dim J(fλ) via a quasiconformal conjugacy, yielding all d by the intermediate value theorem . The model follows the high-level plan but makes key errors: (i) it identifies the maximal pole multiplicity with the odd parameter m from the arcsin step and asserts the lower bound dim J≥2m/(m+1), whereas the paper’s bound is 2q/(q+1) with q=4p, giving 8p/(4p+1) and independent of that odd m ; (ii) it claims all poles have multiplicity m, but in the paper the poles of H and hm have multiplicity 4p . These mistakes undermine the model’s parameter-control argument (it varies m, not p) even though the overall strategy mirrors the paper’s proof.