We prove that a singular-hyperbolic
attractor of a -dimensional
flow is chaotic, in two different strong senses. First, the flow is expansive:
if two points remain close at all times, possibly with time reparametrization,
then their orbits coincide. Second, 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 -Gibbs state
and is 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.
In particular these results can be applied
(i) to the flow defined by the Lorenz equations, (ii) to the geometric Lorenz
flows, (iii) to the attractors appearing in the unfolding of certain resonant
double homoclinic loops, (iv) in the unfolding of certain singular cycles and
(v) in some geometrical models which are singular-hyperbolic but of a different
topological type from the geometric Lorenz models. In all these cases the
results show that these attractors are expansive and have physical measures
which are -Gibbs
states.