Sufficient conditions for robustness of attractors

A recent problem in dynamics is to determinate whether an attractor $\Lambda$ of a $C^r$ flow $X$ is $C^r$ robust transitive or not. By {\em attractor} we mean a transitive set to which all positive orbits close to it converge. An attractor is $C^r$ robust transitive (or {\em $C^r$ robust} for short) if it exhibits a neighborhood $U$ such that the set $\cap_{t>0}Y_t(U)$ is transitive for every flow $Y$ $C^r$ close to $X$. We give sufficient conditions for robustness of attractors based on the following definitions. An attractor is {\em singular-hyperbolic} if it has singularities (all hyperbolic) and is partially hyperbolic with volume expanding central direction \cite{MPP}. An attractor is {\em $C^r$ critically-robust} if it exhibits a neighborhood $U$ such that $\cap_{t>0}Y_t(U)$ is in the closure of the closed orbits is every flow $Y$ $C^r$ close to $X$. We show that on compact 3-manifolds all $C^r$ critically-robust singular-hyperbolic attractors with only one singularity are $C^r$ robust.