Sectional-hyperbolic Lyapunov stable sets

Abstract

In hyperbolic dynamics, a well-known result is: every hyperbolic Lyapunov stable set, is attracting; it's natural to wonder if this result is maintained in the sectional-hyperbolic dynamics. This question is still open, although some partial results have been presented. We will prove that all sectional-hyperbolic transitive Lyapunov stable set of codimension one of a vector field \begin{document}$ X $\end{document} over a compact manifold, with unique singularity Lorenz-like, which is of boundary-type, is an attractor of \begin{document}$ X $\end{document} .