AI 中文总结
本文分类ℝⁿ上的仿射Anosov映射，确定其稳定指标谱，分析Anosov可实现轨迹的边界与内部，证明其在不同拓扑下的开性。
AI 中文摘要
对于n≥2，我们对ℝⁿ上的仿射微分同胚f_{A,v}(x)=Ax+v进行分类，这类映射存在完备黎曼度量，使其成为Anosov映射。该度量存在当且仅当A是双曲的，或f_{A,v}无不动点，等价于v∉Im(I-A)。在后一种情况下，当det A>0时，f_{A,v}光滑共轭于一个平移；当det A<0时，光滑共轭于White映射乘以恒等映射。我们还确定了可能的稳定指标：双曲情形下，指标由A的稳定谱决定；无不动点的映射可具有从1到n-1的所有稳定指标的完备Anosov度量。沿1-特征空间，此类度量中，向量范数必沿两条半轨道之一呈指数增长。随后，我们确定仿射参数空间中Anosov可实现轨迹的内部与边界，描述正则漂移参数附近指标谱的相应变化。最后，证明Anosov可实现性在双侧弱C¹_loc拓扑中不具开性，但在双侧强Whitney C¹拓扑中具开性。
英文摘要
For $n\ge2$, 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 $v\notin\operatorname{Im}(I-A)$. 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 $C^1_{\mathrm{loc}}$ topology but is open in the two-sided strong Whitney $C^1$ topology.
