Entire or rational maps with integer multipliers
Résumé
Let OK be the ring of integers of an imaginary quadratic field K. Recently, Ji and Xie proved that every rational map f:ˆC→ˆC of degree d≥2 whose multipliers all lie in OK 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:C→C whose multipliers all lie in OK is a power map or a Chebyshev map.
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