Ruelle’s Zeta Function for Non-Archimedean Rational Maps

The paper proves that, for subhyperbolic maps on Qp, the Cp-valued Ruelle-type zeta function is a quotient of two entire functions by constructing a 0‑nuclear transfer operator on disc-based dynamical charts and identifying the zeta via a trace formula (hyperbolic case: 1/det(I−zL) equals the exponential of the periodic sum; subhyperbolic case: an extra Euler-product correction makes the result a quotient of two entire functions) . The candidate solution, however, asserts a two-operator (0- and 1‑forms) identity and miscomputes the key local trace difference: tr(T0)−tr(T1) equals g(0), not g(0)/(1−α); consequently the model’s claimed trace identity does not reproduce the required 1/(1−((f^n)'(x))^−1) factor, so the proposed determinant quotient does not match the zeta function as defined. The paper’s argument is essentially correct (with minor notational issues, e.g., the domain for the exponent β), while the model’s proof has a critical algebra error and an impossible geometric assumption about excising postcritical points from a neighborhood containing J(f).