Entire or rational maps with integer multipliers
| Autor: | Buff, Xavier, Gauthier, Thomas, Huguin, Valentin, Raissy, Jasmin |
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| Rok vydání: | 2022 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | Let $\mathcal{O}_{K}$ be the ring of integers of an imaginary quadratic field $K$. Recently, Ji and Xie proved that every rational map $f \colon \widehat{\mathbb{C}} \rightarrow \widehat{\mathbb{C}}$ of degree $d \geq 2$ whose multipliers all lie in $\mathcal{O}_{K}$ is a power map, a Chebyshev map or a Latt\`{e}s 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 $f \colon \mathbb{C} \rightarrow \mathbb{C}$ whose multipliers all lie in $\mathcal{O}_{K}$ is a power map or a Chebyshev map. Comment: 8 pages; added the case of entire maps |
| Databáze: | arXiv |
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