DYNAMICAL ZETA FUNCTIONS FOR DIFFERENTIABLE PARABOLIC MAPS OF THE INTERVAL

The paper proves meromorphic extension of ζ_{T,v} by inducing on the expanding branch, relating ζ_{T,v} to flat determinants of the induced map’s transfer operators, and controlling spectra via a B_{p,q}-space Lasota–Yorke scheme. The candidate solution follows the same high-level plan but implements a standard C^s Lasota–Yorke/Hennion approach and a (sharp/Fredholm-type) determinant ratio. Both reach the same classification of zeros/poles and the same target domain {Λ(z) < ρ^{r−2}} when k ≥ r−2. The model explicitly notes the regularity dependence (k vs. r) and provides a cautious fallback domain {Λ(z) < ρ^{min(k, r−2)}}; the paper states the domain with r−2 but its proof, via Proposition 4.4, implicitly optimizes over k up to r−1. Aside from this regularity bookkeeping, the arguments agree on substance.