Star flows with singularities of different indices

Abstract: It is known that a generic star vector field $X$ on a $3$ or $4$-dimensional manifold is such that its chain recurrence classes are either hyperbolic, or singular hyperbolic ([MPP] and [GSW]). Palis conjectured that every vector field must be approximated either by singular hyperbolic vector fields or by vector fields with homoclinic tangencies or heterodimensional cycles (associated to periodic orbits). We give a counter example in dimension $5$ (and higher). We present here an open set of star vector fields on a $5$-dimensional manifold for which two singular points with different indices belong (robustly) to the same chain recurrence class. This prevents the class to be singular hyperbolic, showing that the results in [MPP] can not be extended to higher dimensions and thus contradicting the conjecture by Palis.