Dessins d'Enfants for Single-Cycle Belyi Maps

TL;DR

This paper explores the connection between Belyi maps and dessins d'enfants, providing explicit descriptions for two infinite families of dynamical Belyi maps, thereby enriching the understanding of their combinatorial and algebraic structures.

Contribution

It completes the classification by explicitly describing dessins d'enfants for two previously studied families of dynamical Belyi maps.

Findings

Explicit dessins d'enfants for two infinite families of Belyi maps.

Enhanced understanding of the combinatorial structures of these maps.

Strengthened link between Belyi maps and dessins d'enfants.

Abstract

Riemann's Existence Theorem gives the following bijections: (1) Isomorphism classes of Belyi maps of degree $d$. (2) Equivalence classes of generating systems of degree $d$. (3) Isomorphism classes of dessins d'enfants with $d$ edges. In previous work, the first author and collaborators exploited the correspondence between Belyi maps and their generating systems to provide explicit equations for two infinite families of dynamical Belyi maps. We complete this picture by describing the dessins d'enfants for these two families.