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

Abstract

AbstractFor 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 ζR(s), which counts the prepenodic orbib of R, according to the weight function |R'| : J(R) → C. An analysis of the analytic domain of ζR(s), using techniques from symbolic dynamics, yields weighted asymptotic formulae for the preperiodic orbits of R. We describe an application to diophantine number theory.