Abstract
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 C 1 + 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 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.
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