Sectional-hyperbolic lyapunov stable sets

TL;DR

This paper investigates whether hyperbolic Lyapunov stable sets are attracting in sectional-hyperbolic dynamics, proving that under certain conditions, such sets are indeed attractors.

Contribution

It establishes that sectional-hyperbolic transitive Lyapunov stable sets of codimension one with a Lorenz-like singularity are attractors.

Findings

Sectional-hyperbolic Lyapunov stable sets of codimension one are attractors.

Proves the result for sets with a Lorenz-like singularity of boundary type.

Extends understanding of stability and attractors in sectional-hyperbolic dynamics.

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 X over a compact manifold, with unique singularity Lorenz-like, which is of boundary-type, is an attractor of X.