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On the volume of singular-hyperbolic sets

J. F. Alves·V. Araujo·M. J. Pacifico·V. Pinheiro
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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 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.

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J. F. Alves, V. Araujo, M. J. Pacifico, V. Pinheiro. 2006-09-11. On the volume of singular-hyperbolic sets. https://arxiv.org/abs/math/0509306

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