THERMODYNAMIC FORMALISM AND HYPERBOLIC BAKER DOMAINS II: REAL-ANALYTICITY OF THE HAUSDORFF DIMENSION

The paper rigorously proves that c ↦ HD(Jr(Fc)) is real-analytic for the Bergweiler family by projecting to the cylinder, building a parameter-holomorphic transfer operator on a fixed Banach space via quasi-conformal conjugacies, applying analytic perturbation theory for the simple leading eigenvalue, and then the Implicit Function Theorem using ∂tP < 0 (Theorem 5.3). The proof depends on detailed prerequisites (expansivity, Bowen’s formula for Jr, Hölder-regular parameter conjugacies) and a technical Proposition verifying analyticity of the operator family, all supplied in the text. See Theorem 5.3 and Proposition 5.2 for the core argument, together with Theorem 3.7 (Bowen’s formula) and the earlier dynamical setup on Q (including Gc via H) and E-hyperbolicity (; ; ; ; ; ; ; ). In contrast, the model’s solution outlines a plausible IFS/inducing approach but does not verify several essential hypotheses (existence and uniformity of a first-return inducing scheme on the right half-cylinder, open set condition/disjointness of branch images, uniform contraction and distortion across parameters, identification of the IFS limit set with Jr, and operator-analytic dependence). It also asserts strict negativity of ∂tP and applicability of analytic IFS perturbation theory without checking summability/space issues. As written, the model’s argument is an incomplete sketch rather than a complete proof.