Mathematics – Dynamical Systems
Scientific paper
2005-09-14
Dynamical Systems, Volume 22, Issue 3, 2007, pages 249 - 267
Mathematics
Dynamical Systems
19 pages, 3 figures; references updated and minor corrections
Scientific paper
An attractor $\Lambda$ 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 $C^{1+\alpha}$ singular-hyperbolic attractors, for some $\alpha>0$, 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 $\Lambda$ 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.
Alves Jose F.
Araujo Vitor
Pacifico Maria Jose
Pinheiro Vilton
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