On the Partial Hyperbolicity of Robustly Transitive Sets with Singularities

Homoclinic tangencies and singular hyperbolicity are involved in the Palis conjecture for vector fields. Typical three dimensional fields well understood by recent works. We study dynamics of higher that away from homoclinic tangencies. More precisely, we prove any field is tangencies, all singularities contained its robustly transitive set hyperbolic have same index. Moreover, $$C^1$$ -generically partially if cannot be accumulated ones with a tangency.