Linearization in ultrametric dynamics in fields of characteristic zero - equal characteristic case

TL;DR

This paper establishes bounds for the size of linearization disks in ultrametric fields of characteristic zero, providing maximal estimates that are sharp in certain cases, extending previous results in p-adic and prime characteristic settings.

Contribution

It offers new bounds for linearization disks in characteristic zero ultrametric fields, filling a gap in the existing literature and generalizing prior p-adic and prime characteristic results.

Findings

Derived bounds are maximal and exact in some cases

Extended previous p-adic and prime characteristic estimates to characteristic zero

Provided examples demonstrating the sharpness of bounds

Abstract

Let $K$ be a complete ultrametric field of charactersitic zero whose corresponding residue field $\Bbbk$ is also of charactersitic zero. We give lower and upper bounds for the size of linearization disks for power series over $K$ near an indifferent fixed point. These estimates are maximal in the sense that there exist exemples where these estimates give the exact size of the corresponding linearization disc. Similar estimates in the remaning cases, i.e. the cases in which $K$ is either a $p$-adic field or a field of prime characteristic, were obtained in various papers on the $p$-adic case (Ben-Menahem:1988,Thiran/EtAL:1989,Pettigrew/Roberts/Vivaldi:2001,Khrennikov:2001) later generalized in (Lindahl:2009 arXiv:0910.3312), and in (Lindahl:2004 http://iopscience.iop.org/0951-7715/17/3/001/,Lindahl:2010Contemp. Math) concerning the prime characteristic case.