B\"ottcher 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 $r\in\mathbb{Z}_{\geq0}$, and let $\phi(x)=x+O(x^{2})$ be the B\"ottcher coordinate satisfying $\phi(g(x))=\phi(x)^{p^{2}}$. Salerno and Silverman conjectured that the radius of convergence of $\phi^{-1}(x)$ 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.