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Let \(\mathcal {O}_{K}\) be the ring of integers of an imaginary quadratic field K. Recently, Ji and Xie proved that every rational map \(f :\widehat{\mathbb {C}} \rightarrow \widehat{\mathbb {C}}\) 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 \(f :\mathbb {C} \rightarrow \mathbb {C}\) whose multipliers all lie in \(\mathcal {O}_{K}\) is a power map or a Chebyshev map.

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Entire or Rational Maps with Integer Multipliers

  • 会议论文

  • 发表日期：

    2026年

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  • Xavier Buff,
  • Thomas Gauthier,
  • Valentin Huguin,
  • Jasmin Raissy

摘要

Let \(\mathcal {O}_{K}\) be the ring of integers of an imaginary quadratic field K. Recently, Ji and Xie proved that every rational map \(f :\widehat{\mathbb {C}} \rightarrow \widehat{\mathbb {C}}\) 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 \(f :\mathbb {C} \rightarrow \mathbb {C}\) whose multipliers all lie in \(\mathcal {O}_{K}\) is a power map or a Chebyshev map.