The Schwarzian derivative and polynomial iteration

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