The Schwarzian derivative and polynomial iteration

Abstract

We consider the Schwarzian derivative S f S_f of a complex polynomial f f and its iterates. We show that the sequence S f n / d 2 n S_{f^n}/d^{2n} converges to − 2 ( ∂ G f ) 2 -2(\partial G_f)^2 , for G f G_f the escape-rate function of f f . As a quadratic differential, the Schwarzian derivative S f n S_{f^n} determines a conformal metric on the plane. We study the ultralimit of these metric spaces.