The Schwarzian derivative and polynomial iteration

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