Singular and sectional hyperbolic sets are the objects of the extension of the classical Smale Hyperbolic Theory to flows having invariant sets with singularities accumulated by regular orbits within the set. It is by now well-known that (partially) hyperbolic sets admit adapted metrics. We show the existence of singular adapted metrics for any singular hyperbolic set with respect to a $C^{1}$ vector field on finite dimensional compact manifolds. Moreover, we obtain 2-sectional adapted metrics for certain open classes of 2-sectional hyperbolic sets and also for any hyperbolic set.

中文翻译：

---

奇异双曲流的自适应度量

奇异双曲集和截面双曲集是经典 Smale 双曲理论扩展到具有不变集的流的对象，其中奇点由集合内的规则轨道累积。众所周知，（部分）双曲线集允许自适应度量。我们展示了关于有限维紧凑流形上的 $C^{1}$ 向量场的任何奇异双曲集的奇异自适应度量的存在。此外，我们为某些 2 截面双曲线集的开放类以及任何双曲线集获得了 2 截面自适应度量。