Preprint A420/2005
Singular hyperbolic attractors are chaotic
Marcelo Viana | Araujo, Vitor | Pacifico, Maria Jose | Pujals, Enrique R.
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.