On the existence of attractors

Abstract:

On every compact $3$-manifold, we build a non-empty open set $\mathcal U$ of $\operatorname {Diff}^1(M)$ such that, for every $r\geq 1$, every $C^r$-generic diffeomorphism $f\in \mathcal U\cap \operatorname {Diff}^r(M)$ has no topological attractors. On higher-dimensional manifolds, one may require that $f$ has neither topological attractors nor topological repellers. Our examples have finitely many quasi-attractors. For flows, we may require that these quasi-attractors contain singular points. Finally we discuss alternative definitions of attractors which may be better adapted to generic dynamics.