Lyapunov stability and sectional-hyperbolicity for higher-dimensional   flows
  We study $C^1$-generic vector fields on closed manifolds without points accumulated by periodic orbits of different indices and prove that they exhibit finitely many sinks and sectional-hyperbolic transitive Lyapunov stable sets with residual basin of attraction. This represents a partial positive answer to conjectures in \cite{am}, the Palis conjecture \cite{pa} and extend the Araujo's thesis to higher dimensions \cite{a}. 
A. Arbieto, C. A. Morales, B. Santiago
math.DS