On the Volume of Singular-hyperbolic Sets
نویسنده
چکیده
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 singularhyperbolic 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 singular-hyperbolic attractors with positive volume. In addition, we show that C generically we have volume zero for C robust classes of singular-hyperbolic attractors.
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