Schwarzian Versus a Family of Moving Parabolic Points

Abstract

In this article, the trajectories of meromorphic quadratic differentials whose coefficients are Schwarzian derivatives of rational maps are studied. We conjecture that the graphs consisting of the so-called negative-real critical Schwarzian trajectories are finite for all rational maps of degree at least two. For any given rational map f, we construct a family of parabolic rational maps { g w } w having a parabolic fixed point at the non-critical point w of f. If w is a zero of the Schwarzian Sf , we prove that the orientations of the negative-real critical trajectories of S f ( z ) d z 2 landing at w coincide with the attracting directions of gw at w. The critical curve in every immediate parabolic basin of w is an arc starting at w along the attracting direction and ending at a critical point of f, which is a pole of Sf . We conduct some numerical experiments to compare the critical curves in the immediate parabolic basins with the negative-real critical Schwarzian trajectories, which support our conjecture.