A vector field $X$ is called a star flow if every periodic orbit of any vector field $C^1$-close to $X$ is hyperbolic. It is known that the chain recurrence classes of a generic star flow $X$ on a 3- or 4-manifold are either hyperbolic, or singular hyperbolic (see [MPP] for 3-manifolds and [LGW] for 4-manifolds).

As it is defined, the notion of singular hyperbolicity forces the singularities in the same class to have the same index. However in higher dimensions (i.e. $≥ 5$), [dL1] shows that singularities of different indices may be robustly in the same chain recurrence class of a star flow. Therefore the usual notion of singular hyperbolicity is not enough for characterizing the star flows.

We present a form of hyperbolicity (called multisingular hyperbolicity) which makes the hyperbolic structure of regular orbits compatible with the one of singularities even if they have different indices. We show that multisingular hyperbolicity implies that the flow is star, and conversely we prove that there is a $C^1$-open and dense subset of the open set of star flows which are multisingular hyperbolic.

More generally, for most of the hyperbolic structures (dominated splitting, partial hyperbolicity etc.) well defined on regular orbits, we propose a way of generalizing it to a compact set containing singular points.

如果任何向量场 $C^1$-接近 $X$ 的每个周期轨道都是双曲线的，则向量场 $X$ 被称为星流。众所周知，3 或 4 流形上的通用星流 $X$ 的链递归类要么是双曲线的，要么是奇异双曲线的（3 流形见 [MPP]，4 流形见 [LGW]）。

正如定义的那样，奇异双曲线的概念迫使同一类中的奇异点具有相同的索引。然而，在更高维度（即 $≥ 5 $）中，[dL1] 表明不同指数的奇点可能在星流的同一链递归类中是稳健的。因此，奇异双曲线的通常概念不足以表征星流。

我们提出了一种双曲线形式（称为多奇异双曲线），它使规则轨道的双曲线结构与奇点之一兼容，即使它们具有不同的指数。我们证明多奇异双曲性意味着流是星形的，相反我们证明了星形流的开集有一个 $C^1$-开且稠密的子集，它是多奇异双曲的。

更一般地，对于在规则轨道上很好定义的大多数双曲结构（支配分裂、部分双曲等），我们提出了一种将其推广到包含奇异点的紧凑集的方法。