On hyperbolic fixed points in ultrametric dynamics

TL;DR

This paper investigates the size of linearization discs near hyperbolic fixed points in ultrametric dynamics, providing bounds that are sharp and characterizing the maximal injectivity region at repelling points.

Contribution

It establishes precise bounds for linearization discs in ultrametric fields and demonstrates their maximality with explicit examples.

Findings

Bounds for linearization disc sizes are sharp and exact in certain cases.

At repelling fixed points, the linearization disc coincides with the maximal injectivity domain.

The results apply to power series over complete ultrametric fields.

Abstract

Let K be a complete ultrametric field. We give lower and upper bounds for the size of linearization discs for power series over K near hyperbolic fixed points. These estimates are maximal in the sense that there exist examples where these estimates give the exact size of the corresponding linearization disc. In particular, at repelling fixed points, the linearization disc is equal to the maximal disc on which the power series is injective.