Entire or rational maps with integer multipliers

Let ${\mathcal{O}}_K$ be the ring of integers of an imaginary quadratic field $K$. Recently, Ji and Xie proved that every rational map $f\text{\hspace{0.17em}⁣}:\widehat{\mathbb{C}}\to \widehat{\mathbb{C}}$ of degree $d\ge 2$ whose multipliers all lie in ${\mathcal{O}}_K$ is a power map, a Chebyshev map or a Lattès map. Their proof relies on a result from non-Archimedean dynamics obtained by Rivera-Letelier. In the present note, we show that one can avoid using this result by considering a differential equation instead. Our proof of Ji and Xie’s result also applies to the case of entire maps. Thus, we also show that every nonaffine entire map $f\text{\hspace{0.17em}⁣}:\mathbb{C}\to \mathbb{C}$ whose multipliers all lie in ${\mathcal{O}}_K$ is a power map or a Chebyshev map.