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.