ZEROS OF DYNAMICAL ZETA FUNCTIONS FOR HYPERBOLIC QUADRATIC MAPS

The paper proves that for f(z)=z^2+c with c<−3.75, if Re s>1/2+ε and |Im s| is large, then L_s u=u forces u≡0, hence Z(s)=det(I−L_s) has only finitely many zeros in that half-plane (Theorem 1.1 and Corollary 1.2) . The proof introduces a refined transfer operator L_{Z,s} (so that L_s u=u ⇒ L_{Z,s}u=u) and uses a semiclassical TT* argument with a phase-separation estimate (Proposition 2.5) and integration by parts to control off-diagonal terms (Section 5; decomposition (5.3)–(5.4) and bounds like (5.14)) . By contrast, the model’s solution relies on a topological-pressure bound derived from the claim sup|g'|≤1/2 to make P(2σ)<0 for σ>1/2 and then applies a schematic Dolgopyat/van der Corput step to all off-diagonal pairs. It does not justify a uniform separation for all word pairs (the paper proves this carefully and even needs c<−3.75 to secure it via (2.12)) , and it also contains an incorrect bound relating φ_α and g'_α. The paper’s argument is complete; the model’s is materially incomplete and contains errors.