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. We prove that these flows exhibit finitely many sinks and sectional-hyperbolic transitive Lyapunov stable sets whose basins form a residual subset of the ambient manifold. This represents a partial positive answer to conjectures in Arbieto and Morales (Proc Am Math Soc 141:2817-2827, ), the Palis conjecture Palis (Nonlinearity 21:T37-T43, ) and gives a flow version of Crovisier and Pujals (Essential hyperbolicity and homoclinic bifurcations: a dichotomy phenomenon/mechanism for diffeomorphisms, ). [ABSTRACT FROM AUTHOR]