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On the volume of singular-hyperbolic sets

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An 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 C1 + α singular-hyperbolic attractors, for any α > 0, always have zero volume, 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 C1 singular-hyperbolic attractors with positive volume. In addition, we show that C1 generically we have volume zero for C1 robust classes of singular-hyperbolic attractors.
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Document Type: Research Article

Affiliations: 1: Centro de Matemática da, Universidade do Porto Rua do Campo, 4169-007 Porto, Portugal 2: Centro de Matemática da, Universidade do Porto Rua do Campo, 4169-007 Porto, Portugal,Instituto de Matemática, Universidade Federal do Rio de Janeiro, RJ-Brazil 3: Instituto de Matemática, Universidade Federal do Rio de Janeiro, RJ-Brazil 4: Departamento de Matemática, Universidade Federal da Bahia, 40170-110 Salvador, Brazil

Publication date: September 1, 2007

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