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Sufficient conditions for robustness of attractors | | C. A. Morales
; M. J. Pacifico
; | | Date: |
25 Mar 2003 | | Subject: | Dynamical Systems MSC-class: Primary 37D30, Secondary 37D45 | math.DS | | Abstract: | 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. | | Source: | arXiv, math.DS/0303310 | | Services: | Forum | Review | PDF | Favorites | |
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