On the existence of attractors

2009/04/28 by Christian Bonatti, Ming Li, Bonatti, Christian +3
Mathematics · #37C05 #37C20 #37C25 #37C29 #37D30 #Dynamical Systems (math.DS) #FOS: Mathematics #math.DS #msc:37C05 #msc:37C20 #msc:37C25 #msc:37C29 #msc:37D30

paper · pdf · doi:10.48550/arxiv.0904.4393

18 pages

arxiv created 2009/04/28 · arxiv updated 2009/12/01

Abstract

On every compact 3-manifold, we build a non-empty open set \cU of \Diff¹(M) such that, for every r≥ 1, every C^r-generic diffeomorphism f∈\cU∩ \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.

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