Preprint D13/2005
On the volume of singular-hyperbolic sets

Vilton Pinheiro | Alves, Jose | Araujo, Vitor | Pacifico, Maria JosÃ©

**Keywords: **
singular-hyperbolic set | partial hyperbolicity | transitive Anosov flow

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.