Key points are not available for this paper at this time.
Morales, Pacifico and Pujals proved recently that every robustly transitive singular set for a3-dimensional flow must be partially hyperbolic. In this paper wegeneralize the resultto higher dimensions. By definition, anisolated invariant set of a C¹ vector field S iscalled robustly transitive if there exist an isolatingneighborhood U of in M and a C¹ neighborhood Uof S in X¹ (M) such that for every X U, the maximal invariantset of X in U is non-trivially transitive. Such a set is called singular if it containsa singularity. The set is called strongly homogeneous of index i, if there exist an isolatingneighborhood U of in M and a C¹ neighborhood Uof S in X¹ (M) such that all periodic orbits ofall X U contained in U havethe same index i. We prove in this paper that any robustlytransitive singular setthat is strongly homogeneous must be partially hyperbolic, as longas the indices of singularities and periodic orbits fit in certainway. As corollaries we obtain that every robustsingular attractor (or repeller) that is strongly homogeneous must bepartially hyperbolic and, if dimM 4, every robustlytransitive singular set that is strongly homogeneous must bepartially hyperbolic. The main novelty of the proofs in this paper is anextension of the usual linear Poincaré flow 'to singularities'.
Li et al. (Sat,) studied this question.