On the volume of singular-hyperbolic sets

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dc.creatorAlves, J. F.
dc.creatorAraujo, V.
dc.creatorPacifico, M. J.
dc.creatorPinheiro, V.
dc.date2005-09-14
dc.date2006-09-11
dc.date.accessioned2026-07-07T08:41:59Z
dc.date.available2026-07-07T08:41:59Z
dc.descriptionAn attractor $Λ$ for a 3-vector field $X$ is singular-hyperbolic if all its singularities are hyperbolic and it is partially hyperbolic with volume expanding central direction. We prove that $C^{1+α}$ singular-hyperbolic attractors, for some $α>0$, always have zero volume, thus extending an analogous result for uniformly hyperbolic attractors. The same result holds for a class of higher dimensional singular attractors. Moreover, we prove that if $Λ$ is a singular-hyperbolic attractor for $X$ then either it has zero volume or $X$ is an Anosov flow. We also present examples of $C^1$ singular-hyperbolic attractors with positive volume. In addition, we show that $C^1$ generically we have volume zero for $C^1$ robust classes of singular-hyperbolic attractors.
dc.description19 pages, 3 figures; references updated and minor corrections
dc.identifierhttps://arxiv.org/abs/math/0509306
dc.identifierhttp://arxiv.org/abs/math/0509306
dc.identifierDynamical Systems, Volume 22, Issue 3, 2007, pages 249 - 267
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/141848
dc.subjectDynamical Systems
dc.subject37D30, 37F45, 37C10, 37C20
dc.titleOn the volume of singular-hyperbolic sets
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