On the volume of singular-hyperbolic sets

Alves, J. F.; Araujo, V.; Pacifico, M. J.; Pinheiro, V.

On the volume of singular-hyperbolic sets

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Description

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.
19 pages, 3 figures; references updated and minor corrections