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}.