Böttcher coordinates at wild superattracting fixed points
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866929272615075840 |
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| author | Fu, Hang Nie, Hongming |
| author_facet | Fu, Hang Nie, Hongming |
| contents | Let $p$ be a prime number, let $g(x)=x^{p^{2}}+p^{r+2}x^{p^{2}+1}$ with $r\in\mathbb{Z}_{\geq0}$, and let $ϕ(x)=x+O(x^{2})$ be the Böttcher coordinate satisfying $ϕ(g(x))=ϕ(x)^{p^{2}}$. Salerno and Silverman conjectured that the radius of convergence of $ϕ^{-1}(x)$ in $\mathbb{C}_{p}$ is $p^{-p^{-r}/(p-1)}$. In this article, we confirm that this conjecture is true by showing that it is a special case of our more general result. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_07867 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Böttcher coordinates at wild superattracting fixed points Fu, Hang Nie, Hongming Dynamical Systems Number Theory Let $p$ be a prime number, let $g(x)=x^{p^{2}}+p^{r+2}x^{p^{2}+1}$ with $r\in\mathbb{Z}_{\geq0}$, and let $ϕ(x)=x+O(x^{2})$ be the Böttcher coordinate satisfying $ϕ(g(x))=ϕ(x)^{p^{2}}$. Salerno and Silverman conjectured that the radius of convergence of $ϕ^{-1}(x)$ in $\mathbb{C}_{p}$ is $p^{-p^{-r}/(p-1)}$. In this article, we confirm that this conjecture is true by showing that it is a special case of our more general result. |
| title | Böttcher coordinates at wild superattracting fixed points |
| topic | Dynamical Systems Number Theory |
| url | https://arxiv.org/abs/2304.07867 |