= Coleman norm-fixed lift
{c}
{title2=$w(\overline f)=\lim_{n\to\infty}N^nf$}
Every <unit> $\overline f\in\mathbb F_p[[T]]^\times$ has a unique <Coleman norm operator>-fixed lift to $\mathbb Z_p[[T]]^\times$. Choose any lift $f$ and use $Nf\equiv f\pmod p$. The consecutive quotients of the iterates are $N^n(Nf/f)\in1+p^{n+1}R$ by <Coleman norm contraction>, so they converge coefficientwise uniformly in the <p-adic integers>. Their limit is norm-fixed by continuity. Two such lifts have norm-fixed quotient in $1+pR$ and therefore coincide. Uniqueness also makes the lifting map multiplicative and independent of the initial lift.
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