On Araujo's Theorem for flows

TL;DR

This paper extends Araujo's theorem from surface diffeomorphisms to three-dimensional flows, showing that generic $C^1$ vector fields without singularities have either infinitely many sinks or finitely many hyperbolic attractors with full measure basins.

Contribution

It generalizes Araujo's result from surface diffeomorphisms to three-dimensional flows, establishing similar dichotomies for generic $C^1$ vector fields.

Findings

Generic three-dimensional flows without singularities have either infinitely many sinks or finitely many hyperbolic attractors.

The basins of these hyperbolic attractors have full Lebesgue measure.

The result extends the understanding of dynamical behavior from surfaces to three-dimensional manifolds.

Abstract

Araujo proved in his thesis \cite{A} that a $C^1$ generic surface diffeomorphism has either infinitely many sinks (i.e. attracting periodic orbits) or finitely many hyperbolic attractors with full Lebesgue measure basin. The goal of this paper is to extend this result to $C^1$ vector fields on compact connected boundaryless manifolds $M$ of dimension 3 (three-dimensional flows for short). More precisely, we shall prove that a $C^1$ generic three-dimensional flow without singularities has either infinitely many sinks or finitely many hyperbolic attractors with full Lebesgue measure basin.