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Sufficient conditions for robustness of attractors

C. A. Morales·M. J. Pacifico
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Abstract

A recent problem in dynamics is to determinate whether an attractor $Λ$ 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.

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C. A. Morales, M. J. Pacifico. 2003-03-25. Sufficient conditions for robustness of attractors. https://arxiv.org/abs/math/0303310

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