Lyapunov stable chain recurrence classes for singular flows

TL;DR

This paper proves that for generic vector fields away from homoclinic tangencies, Lyapunov stable chain recurrence classes are homoclinic classes, using an approach involving convergence of Gibbs states.

Contribution

It establishes a new characterization of Lyapunov stable chain recurrence classes as homoclinic classes for generic singular flows.

Findings

Lyapunov stable classes are homoclinic classes for generic fields

Use of Gibbs $F$-states convergence in the proof

Applicable to $C^1$ generic vector fields away from tangencies

Abstract

We show that for a $C^1$ generic vector field $X$ away from homoclinic tangencies, a nontrivial Lyapunov stable chain recurrence class is a homoclinic class. The proof uses an argument with $C^2$ vector fields approaching $X$ in $C^1$ topology, with their Gibbs $F$-states converging to a Gibbs $F$-state of $X$.