On Araujo's Theorem for flows

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.