Decompletion of cyclotomic perfectoid fields in positive characteristic

Let $E$ be a field of characteristic $p$. The group ${\mathbf{Z}}_p^{\times }$ acts on $E((X))$ by $a\cdot f(X)=f((1+X{)}^a-1)$. This action extends to the $X$-adic completion $\tilde{\mathbf{E}}$ of ${\cup }_{n\ge 0}E((X^{1\mathrm{/}{p}^n}))$. We show how to recover $E((X))$ from the valued $E$-vector space $\tilde{\mathbf{E}}$ endowed with its action of ${\mathbf{Z}}_p^{\times }$. To do this, we introduce the notion of super-H"older vector in certain $E$-linear representations of ${\mathbf{Z}}_p$. This is a characteristic $p$ analogue of the notion of locally analytic vector in $p$-adic Banach representations of $p$-adic Lie groups.