Preprint A420/2005
Singular hyperbolic attractors are chaotic

Marcelo Viana | Araujo, Vitor | Pacifico, Maria Jose | Pujals, Enrique R.

**Keywords: **

We prove that a singularhyperbolic (or Lorenzlike) attractor of a 3dimensional
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 SinaiRuelleBowen) 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 centerunstable direction,
is a uGibbs state and an equilibrium state for the logarithm of the Jacobian of the time one map
of the flow along the strongunstable direction. In particular these results show that both the flow
defined by the Lorenz equations and the geometric Lorenz flows are expansive and have physical
measures which are uGibbs states.