One-parameter families of Schwarz reflection maps arising from Shabat-Belyi maps
Published in Stony Brook University (PhD Thesis), 2024
Recommended citation: J. Mazor. (2024). "One-parameter families of Schwarz reflection maps arising from Shabat-Belyi maps." PhD Thesis, Stony Brook University. https://www.math.stonybrook.edu/alumni/2024-Jacob-Mazor.pdf
We study one-parameter families of Schwarz reflection maps of the Riemann sphere whose critical values are determined by Shabat-Belyi maps. These families arise naturally at the interface of holomorphic dynamics and the theory of dessins d’enfants. We analyze the structure of the parameter spaces, the geometry of the associated limit sets, and the relationship between Schwarz reflection maps and antiholomorphic correspondences, contributing to the Fatou-Sullivan dictionary between rational maps and Kleinian groups.