Dynamics of convergent power series on the integral ring of a finite extension of $\Qp$

TL;DR

This paper investigates the dynamics of convergent power series over the integral ring of a finite extension of p-adic numbers, providing a detailed minimal decomposition and revealing the existence of uncountably many minimal subsystems.

Contribution

It establishes a minimal decomposition theorem for these dynamical systems and characterizes all affine systems' minimal decompositions.

Findings

Existence of uncountably many minimal subsystems

Complete minimal decompositions for affine systems

Conditions for minimal sets with infinitely many points

Abstract

Let $K$ be a finite extension of the field $\mathbb{Q}_p$ of $p$-adic numbers and $\O$ be its integral ring. The convergent power series with coefficients in $\O$ are studied as dynamical systems on $\O$. A minimal decomposition theorem for such a dynamical system is obtained. It is proved that there are uncountably many minimal subsystems, provided that there is a minimal set consisting of infinitely many points. In particular, the complete detailed minimal decompositions of all affine systems are derived.