Affine Anosov Maps on ${\mathbb{R}}^n$: Classification, Index Spectrum, and Stability at Infinity

For $n\ge 2$, we classify the affine diffeomorphisms $f_{A,v}(x)=Ax+v$ on ${\mathbb{R}}^n$ that admit a complete Riemannian metric with respect to which they are Anosov. Such a metric exists if and only if $A$ is hyperbolic or $f_{A,v}$ has no fixed point, equivalently $v\mathrm{\notin }\mathrm{Im}(I-A)$. In the latter case, $f_{A,v}$ is smoothly conjugate to a translation when $\det A>0$ and to White’s map times the identity when $\det A<0$. We also determine the possible stable indices. In the hyperbolic case, the index is determined by the stable spectrum of $A$, whereas a map with no fixed point admits complete Anosov metrics of every stable index from $1$ to $n-1$. Along the $1$-eigenspace, every such metric must exhibit exponential growth of vector norms along one of the two half orbits. We then determine the interior and boundary of the Anosov-realizable locus in the affine parameter space and describe the corresponding change of the index spectrum near regular drift parameters. Finally, we show that Anosov-realizability is not open in the two-sided weak $C_{\mathrm{l}\mathrm{o}\mathrm{c}}^1$ topology but is open in the two-sided strong Whitney $C^1$ topology.