  We prove that a singular-hyperbolic attractor of a 3-dimensional flow is
chaotic, in two strong different senses. Firstly, the flow is expansive: if two
points remain close for all times, possibly with time reparametrization, then
their orbits coincide. Secondly, there exists a physical (or
Sinai-Ruelle-Bowen) measure supported on the attractor whose ergodic basin
covers a full Lebesgue (volume) measure subset of the topological basin of
attraction. Moreover this measure has absolutely continuous conditional
measures along the center-unstable direction, is a $u$-Gibbs state and an
equilibrium state for the logarithm of the Jacobian of the time one map of the
flow along the strong-unstable direction. This extends to the class of
singular-hyperbolic attractors the main elements of the ergodic theory of
uniformly hyperbolic (or Axiom A) attractors for flows.
