Böttcher coordinates at wild superattracting fixed points

Abstract

Let $p$ be a prime number, let $g(x)=x^{p^2}+p^{r+2}{x}^{p^2+1}$ with $r\in {\mathbb{Z}}_{\ge 0}$, and let $\ddot{I}•(x)=x+O(x^2)$ be the Böttcher coordinate satisfying $\ddot{I}•(g(x))=\ddot{I}•(x{)}^{p^2}$. Salerno and Silverman conjectured that the radius of convergence of $\ddot{I}{•}^{-1}(x)$ 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.