This 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 $\exp(C_K |s|^{ \delta})$ in strips $|\Re s| \leq K$, where $\delta$ 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 $\{ |\Re s | \leq |\Im s|^\alpha\}$ is given, followed by weaker lower bound estimates in strips $\{ \Re s > -C, |\Im s|\leq r\}$, and logarithmic neighbourhoods $\{ |\Re s | \leq \rho \log |\Im s| \}$. Recent numerical work of Strain-Zworski suggests the upper bounds in strips are optimal.
Categories: math.DS math.CV