Notation for the problem:
Let $E/\mathbb{Q}_P$ be a local field, and $\mu_E$ its maximal ideal. Let $K=E\{\{T\}\}$ be the standard 2-dimensional local field equipped with the Parshin topology and let $\mu_K=\mu_E\{\{T\}\}$
Let $F$ be a formal group with coefficients in $\mathbb{Z}_p$, you may assume is a Lubin-Tate formal group for simplicity.
Let $f=[p]_F$ be the multiplication-by-$p$ map and denote by $\kappa_n=\{z:f(z)=0\}$ the $p$th-torsion points. We will assume that $\kappa_n \subset E$.
Finally, . Let us fix a Galois element $g$ in $G_K^{ab}$, the maximal abelian galois group of $K$.
Let $x\in \mu_K$ and take a $z$ such that $f(z)=x$. Consider the element $g(z)\ominus_Fz$ (this element lies in $\kappa_n$ since both $z$ and g(z) are solutions of $f(Z)=x$).
Question1: if $x\to 0$ in the Parshin topology can we say that eventually $g(z)\ominus_Fz=0$, i.e, if $x$ is close enough to zero then $g(z)=z$?
Question2: If the sequence $x_n\to 0$ then can we find a sequence $z_n$, such that $f(z_n)=x_n$ and $z_n\to 0$?
