Analyticity of the Hausdorff dimension and metric structures on Misiurewicz families of polynomials

The paper’s main claim in this part (Theorem 1.2) is that the path-length function d_G induced by the Hessian 2-form ⟨·,·⟩_G is a distance on any bounded Λ-hyperbolic component Ω of a Misiurewicz subfamily in polycm_D; see the statement of results and Theorem 1.2 (d_G is a distance) in the PDF. The authors emphasize that Misiurewicz dynamics is not uniformly hyperbolic on the Julia set (critical points obstruct uniform expansion) and hence they build a tower and a 4-parameter family of transfer operators to obtain analyticity and the needed thermodynamic control, rather than appealing to the standard Ruelle theory for uniformly expanding repellers . They prove real-analyticity of a joint pressure P(t1,λ1,t2,λ2) via spectral gap on the tower and Kato–Rellich perturbation, recovering analyticity of δ(λ) and of G_{λ0}(λ)=δ(λ)∫log|f'_{λ}∘h_{λ}| dν, and define the Hessian form ⟨·,·⟩_G accordingly . They link the Hessian to a variance-type pressure form and prove that every nontrivial C^1 path has strictly positive length (Proposition 4.2), using Lemma 4.1 plus the constancy of the Lyapunov exponent L(λ)=log D on bounded polynomial components, and then conclude separation via an analyticity/stratification argument in Section 5 . By contrast, the model’s solution incorrectly assumes uniform expansion on J_λ within Ω and concludes that the Hessian is positive definite on moduli (killing only Möbius/affine conjugacy directions). The paper explicitly cautions that non-degeneracy may fail and does not assert pointwise positive definiteness; it instead proves that path lengths are positive and d_G separates points without assuming a uniform lower bound of the form ⟨v,v⟩_G ≥ c‖v‖^2 on small balls . The model’s separation argument (Step 5) hinges on such a local lower bound and therefore does not apply in this Misiurewicz setting.