Abstract:
For a Belyi function $\beta\colon \mathbb{CP}^1\rightarrow \mathbb{CP}^1$ ramified only over the points $-1$, $1$, and $\infty$, a corresponding «dessin d'enfant»$\mathscr{D}_{\beta}$ 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 the support of $\mathscr{D}_{\beta}$. In this note, we solve the following problem: under what conditions different dessins $\mathscr{D}_{\beta_1}$ and $\mathscr{D}_{\beta_2}$ have equal supports?
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