On dessins d'enfants with equal supports

Abstract

For a Belyi function $\beta:\mathbb C\mathbb P^1\rightarrow \mathbb C\mathbb P^1$ ramified only over the points $-1,1,\infty$, a corresponding ``dessin d'enfant'' $\mathcal D_{\beta}$ is defined as the set $\beta^{-1}([-1,1])$ considered as a bi-colored graph on the Riemann sphere whose white and black vertices are points of the sets $\beta^{-1}\{-1\}$ and $\beta^{-1}\{1\}$ correspondingly. Merely the set $\beta^{-1}([-1,1])$ without a graph structure is called a support of $\mathcal D_{\beta}$. In this note, we solve the following problem: under what conditions different dessins $\mathcal D_{\beta_1}$ and $\mathcal D_{\beta_2}$ have equal supports?