Topological conjugacy classes of affine maps

TL;DR

This paper classifies affine maps over real and complex fields up to topological conjugacy, showing that fixed points and the linear part's properties determine conjugacy classes in low dimensions.

Contribution

It provides a complete classification of affine maps in dimensions one and two over real and complex fields based on fixed points and linear part conjugacy.

Findings

Affine maps with fixed points are conjugate iff their linear parts are conjugate.

In dimensions 1 and 2 without fixed points, conjugacy depends on singularity of the linear part.

Classification is achieved for affine maps in ^n with n   2.

Abstract

A map $f: \ff^n \to \ff^n$ over a field $\ff$ is called affine if it is of the form $f(x)=Ax+b$, where the matrix $A \in \ff^{n\times n}$ is called the linear part of affine map and $b \in \ff^n$. The affine maps over $\ff=\rr$ or $\cc$ are investigated. We prove that affine maps having fixed points are topologically conjugate if and only if their linear parts are topologically conjugate. If affine maps have no fixed points and $n=1$ or 2, then they are topologically conjugate if and only if their linear parts are either both singular or both non-singular. Thus we obtain classification up to topological conjugacy of affine maps from $\ff^n$ to $\ff^n$, where $\ff=\rr$ or $\cc$, $n\leq 2$.