Star fows and multisingular hyperbolicity

Christian Bonatti, Adriana da Luz

Published 2017-05-16Version 1

A vector field X is called a star flow if every periodic orbit, of any vector field C1-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 [GLW] on 4-manifolds). In higher dimension (i.e at least 5) another phenomena can happen: singularities of different indices may be robustly in the same chain recurrence class of a star flow. We present a form of hyperbolicity (called multi-singular hyperbolic) which makes compatible the hyperbolic structure of regular orbits together with the one of singularities even if they have different indexes. We show that multisingular hyperbolicity implies that the flow is star, and conversely, there is a C1-open and dense subset of the an open set of star flows which are multisingular hyperbolic.