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 ${\left\{g_w\right\}}_w$ having a parabolic fixed point at the non-critical point w of f. If w is a zero of the Schwarzian S_f, we prove that the orientations of the negative-real critical trajectories of $S_f(z)\mathrm{d}{z}^2$ landing at w coincide with the attracting directions of g_w 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 S_f. 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.

Keywords:

• Schwarzian derivatives
• trajectories
• parabolic rational maps

2020 Mathematics Subject Classification:

• Primary 30D30
• Secondary 37F10
• 37F75

Acknowledgments

This article was written based on a group discussion led by Bill Thurston in 2010, and the motivation can be found in Bill’s post on Mathoverflow [8]. We thank to John H. Hubbard, Sarah Koch, Kevin Pilgrim, and Reinhard Schaefke for helpful discussions. We are very grateful to the referee for his/her quick feedback, including encouragements and valuable suggestions on the first version of this article. We also thank to the referee for very careful reading, very helpful suggestions about the writing and organization on the second version of this article. The preparation of this article was sadly overshadowed by the early death of Bill Thurston (1946–2012) and the second author Tan Lei (1963–2016). We dedicate this work to their memory.

Declaration of Interest

No potential conflict of interest was reported by the author(s).

Notes

Notes

¹ We would like to thank the referee for providing such an interesting example.

Additional information

Funding

F. Y. was supported by the NSFC (No. 12071210), NSF of Jiangsu Province (No. BK20191246) and the Fundamental Research Funds for the Central Universities (No. 0203-14380033).