Starflows with singularities of different indices

Adriana da Luz

Published 2018-06-23Version 1

A vector field X is called a star flow if every periodic orbit of any vector field C1-close to X is hypebolic. It is known that a generic star flow in dimensions 3 and 4 are such that the chain recurrence classes are either hyperboilc or singular hyperbolic ([MPP] and [GSW]) We present a nonempty open set of star flows on a 5 dimensional manifold for which two singular points of different indices belong (robusly) to the same chain recurrence class. This prevents the class to be singular hyperbolic. We show that this chain recurrence class is robustly chain transitive. The firts and only example of this phenomena is in [BCGP]. We proove that this example has a weak form of hyperbolicity called strong multisingular hyperbolicity. This is a particular case of the multisingular hyperbolicity presented in [BdL] and this implies that the example is a star flow.