Singular hyperbolicity of $C^1$ generic three dimensional vector fields

TL;DR

This paper proves that for a generic class of three-dimensional vector fields, all isolated transitive sets exhibit singular hyperbolicity, advancing understanding of dynamical systems on manifolds.

Contribution

It provides a partial confirmation of a conjecture by showing singular hyperbolicity for generic $C^1$ vector fields on 3D manifolds.

Findings

All isolated transitive sets are singular hyperbolic for generic $C^1$ vector fields.

Supports the conjecture in extcite{MP} in the three-dimensional case.

Advances the classification of dynamical behaviors in 3D systems.

Abstract

In the paper, we show that for a generic $C^1$ vector field $X$ on a closed three dimensional manifold $M$, any isolated transitive set of $X$ is singular hyperbolic. It is a partial answer of the conjecture in \cite{MP}.