Preprint A288/2004
EXISTENCE OF PERIODIC ORBITS FOR SINGULAR-HYPERBOLIC ATTRACTORS
CARLOS MORALES | BAUTISTA, SERAFIN
Keywords: Partially Hyperbolic Set | Attractor | Flow
An \emph{attractor} is a transitive set to which all nearby positive orbits converge. A non-trivial attractor for flows is \emph{singular-hyperbolic} if it has singularities (all hyperbolic) and is partially hyperbolic with volume expanding central direction. We show that a singular-hyperbolic attractor of a $C^1$ flow on a compact $3$-manifold has a periodic orbit. This result has the following corollaries. First every singular-hyperbolic attractor has topological dimension $\geq 2$ (solving positively a question posed in \cite{m2}). Second any of such attractors is the closure of the unstable manifold of a periodic orbit. Third every $C^1$ robust transitive set has a periodic orbit. Our result generalize well known properties of hyperbolic and geometric Lorenz attractors \cite{pt}.