Topological conjugacy classes of affine maps

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 $\cc$, $n\leq 2$.