Divergence and convergence of conjugacies in non-Archimedean dynamics

TL;DR

This paper investigates the conditions under which power series over non-Archimedean fields are linearizable near indifferent fixed points, revealing new criteria based on the divisibility of monomial degrees and analyzing the size of linearization discs.

Contribution

It establishes that power series with monomials of degrees divisible by p are analytically linearizable and determines bounds for the size of the linearization disc, extending understanding of conjugacy divergence.

Findings

Power series with monomials of degrees divisible by p are linearizable.

Exact size of the linearization disc can be determined in certain cases.

Existence of conjugacy divergence when polynomials contain monomials of degree prime to p.

Abstract

We continue the study in [21] of the linearizability near an indif- ferent fixed point of a power series f, defined over a field of prime characteristic p. It is known since the work of Herman and Yoccoz [13] in 1981 that Siegel's linearization theorem [27] is true also for non- Archimedean fields. However, they also showed that the condition in Siegel's theorem is 'usually' not satisfied over fields of prime character- istic. Indeed, as proven in [21], there exist power series f such that the associated conjugacy function diverges. We prove that if the degrees of the monomials of a power series f are divisible by p, then f is analyt- ically linearizable. We find a lower (sometimes the best) bound of the size of the corresponding linearization disc. In the cases where we find the exact size of the linearization disc, we show, using the Weierstrass degree of the conjugacy, that f has an indifferent periodic point on the boundary. We also give a class of polynomials containing a monomial of degree prime to p, such that the conjugacy diverges.