2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/71012This paper describes new results on the growth and zeros of the Ruelle zeta function for the Julia set of a hyperbolic rational map. It is shown that the zeta function is bounded by $\exp(C_K |s|^δ)$ in strips $|\Re s| \leq K$, where $δ$ is the dimension of the Julia set. This leads to bounds on the number of zeros in strips (interpreted as the Pollicott-Ruelle resonances of this dynamical system). An upper bound on the number of zeros in polynomial regions $\{|\Re s | \leq |\Im s|^α\}$ is given, followed by weaker lower bound estimates in strips $\{\Re s > -C, |\Im s|\leq r\}$, and logarithmic neighbourhoods $\{|\Re s | \leq ρ\log |\Im s| \}$. Recent numerical work of Strain-Zworski suggests the upper bounds in strips are optimal.18 pages, 1 figure Expanded Lemma 5.2 and moved to an appendixDynamical SystemsComplex Variables37C30text