Lyapunov stability and sectional-hyperbolicity for higher-dimensional flows

TL;DR

This paper investigates the stability and hyperbolic properties of higher-dimensional flows, showing that generic vector fields have finitely many sinks and sectional-hyperbolic stable sets, supporting key conjectures in dynamical systems.

Contribution

It extends the understanding of Lyapunov stability and sectional-hyperbolicity to higher dimensions, providing partial answers to longstanding conjectures.

Findings

Finitely many sinks in generic vector fields

Existence of sectional-hyperbolic transitive Lyapunov stable sets

Residual basin of attraction for these sets

Abstract

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}.