ZETA-FUNCTIONS AND ASYMPTOTIC FORMULAS FOR PREPERIODIC ORBITS OF HYPERBOLIC RATIONAL MAPS

For a hyperbolic rational map R of the Riemann sphere of degree d grea ter than or equal to 2, restricted to its Julia set J(R), we define a zeta function zeta(R)(s), which counts the preperiodic orbits of R, ac cording to the weight function \R'\ : J(R) --> C. An analysis of the a nalytic domain of zeta(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.