The Schwarzian derivative and polynomial iteration

TL;DR

This paper investigates the behavior of the Schwarzian derivative under polynomial iteration, showing convergence properties and their implications for conformal metrics and ultralimits in complex dynamics.

Contribution

It establishes the convergence of normalized Schwarzian derivatives of polynomial iterates to a specific quadratic differential related to the escape-rate function.

Findings

Normalized Schwarzian derivatives converge to a quadratic differential

The Schwarzian derivative determines a conformal metric on the plane

Ultralimits of these metrics are studied

Abstract

We consider the Schwarzian derivative $S_f$ of a complex polynomial $f$ and its iterates. We show that the sequence $S_{f^n}/d^{2n}$ converges to $-2(\partial G_f)^2$, 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.