Star flows and multisingular hyperbolicity

Abstract

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. $\ge 5$), [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.