Zeta Functions and Asymptotic Formulae for Preperiodic Orbits of Hyperbolic Rational Maps

For a hyperbolic rational map R of the Riemann sphere of degree d ≥ 2, restricted to its Julia set J(R ), we define a zeta function ζ R ( s ), which counts the prepenodic orbib of R , according to the weight function |R'| : J(R) → C. An analysis of the analytic domain of ζ R ( s ), using techniques from symbolic dynamics, yields weighted asymptotic formulae for the preperiodic orbits of R . We describe an application to diophantine number theory.