Non-Linearizability of power series over complete non-Archimedean fields of positive characteristic

Abstract

In [H-Y83], Herman and Yoccoz prove that for any given locally analytic (at $z=0$) power series $f(z)=z(\lambda +{\sum }_{i=1}^{\infty }{a}_i{z}^i)$ over a complete non-Archimedean field of characteristic $0$ if $|\lambda |=1$ and $\lambda$ is not a root of unity, then $f$ is locally linearizable at $z=0$. They ask the same question for power series over fields 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. More precisely, given a complete non-Archimedean field $\mathcal{K}$ of positive characteristic and a power series $f(z)=z(\lambda +{\sum }_{i=1}^{\infty }{a}_i{z}^i)\in \mathcal{K}[[z]]$ with $\lambda$ not a root of unity and $|1-\lambda |<1$, we prove a sufficient condition (Criterion~\star) for $f$ to be non-linearizable. This phenomenon of prevalence for power series over fields of positive characteristic being non-linearizable was initially conjectured in [Her87, p 147] by Herman, and formulated into a concrete question by Lindahl as [Lin04, Conjecture 2.2]. As applications of our criterion, we prove the non-linearizability of three families of polynomials.