Non-Linearizability of power series over an untrametric field of positive characteristic

In [HY83], Herman and Yoccoz prove that every power series $f(T)=T(\lambda +\sum_{i=1}^\infty a_iT^i) \in \mathbb{Q}_p[\![T]\!]$ such that $|\lambda|=1$ and $\lambda$ is not a root of unity is linearizable. They asked the same question for power series in $\mathcal K[\![T]\!]$, where $\mathcal K$ is an ultrametric field of positive characteristic. In this paper, we prove that, on opposite, most such power series in this case are more likely to be non-linearizable, which was conjectured in [p 147, Her87] by Herman. More precisely, for $f(T)=T(\lambda +\sum\limits_{i=1}^\infty a_iT^i) \in \mathcal K[\![T]\!]$ such that $\lambda$ is not a root of unity and $0<|1-\lambda|<1$, we prove a sufficient condition (Criterion \star) of $f$ to be non-linearizable. By this criterion, we are able to show that a family of cubic polynomials over $\mathcal K$ is non-linearizable.